Big Ideas Math IM 2 11 4 Volumes of Prisms and Cylinders Lecture Problem Set

well and welcome YouTube Mr Robinson back here with you another brand new exciting video all math based of course and as always it is an honor and privilege to be serving you here today as it is every day here in my virtual classroom step on theide as we jump into section 11.4 in the Big Ideas Math integrated math 2 textbook on volumes of prisms and pyramids now everything that we've done from 111 to 113 feel is leading up to everything we do from 114 to 117 or at least the the latter half of the section on the volumes and surface areas and things like that because you need to know how to find areas of things to get those you need to know how circumference works and all that it's just all buildup and those things you might have known in the past from 111 to 113 or at least in some way maybe no things on volumes and if so maybe this video is not for you in which case you can switch off otherwise go and check this one out we got solution VI guide video coming up after the lecture portion if you want to find timestamps for this go to the description section down below you can find all that material there and you can find PDF of this as well so go ahead and check that out we're going to start with that lecture portion right now have calculator handy if you're going to be following along because we're definitely going to be doing lot of rounding now we are going to classify solids that means give them their name just like how in 2D shapes polygons we give square and you know things like that we're going to call things what they are and find volumes of prisms and cylinders that's that's the head point lot of new core vocabulary polyhedron kind of the polygon of the 3D World face kind of like the side of the 3D World Edge is the sides that we know vertex like the angle of the 3D World volume like the area of the 3D World Cavalier Cavalier Cavalier's principle have no idea what that is we'll find out together and similar solids kind of like similar polygons similar lengths and things like that similarity similarity we're going to refer to how things are cubed in that way previous things we'd have to know from before the word solid basically of 3D shape prism well don't remember when we actually used these so I'm going to name them out so solid prism pyramid cylinder cone sphere base and composite solid yeah don't know when those things specifically did pop up maybe they did and I'm just totally forgetful do apologize on that okay let's classify solids hey three-dimensional figure or solid is bounded by the flat or curved surfaces that enclose the single region of space polyhedrin is solid that's bounded by polygons called faces so sorry was going to go back again one more time solid like sphere is solid but it's not polyhedron just like circle is shape guess the 2D word is shape but it's not polygon because it doesn't have like sides as spheres don't have edges so polyhedrin is solid that's bounded by polygons called faces so they has polygons an edge of polyhedrin is line segment boom like that formed by the intersection of two faces and Vertex is point where three or more edges meet so again Edge is kind of like the side of old face is like your new kinds of sides that we'd say here and vertex is like an angle the plural of polyed is polyhedra or polyhedrin tend to say the latter believe okay so several types of polyhedrin yes would say that are prisms and pyramids because you can see all the faces are of polygons these are generally rectangles and maybe that's square base but with triangles at the top we'll be classifying these in general calling them like what makes pyramid versus prism I'm sure they're going to separate those they didn't mention what pyramid is just yet not polyhedrons cylinder is not con's not sphere is not their connection is they contain something that's not polyon as facing the cylinder has two circular bases the cone has circular base the sphere doesn't technically maybe have any faces but it's completely circular in nature so like said if circles are not polygons then that doesn't work okay to name prism or pyramid you use the shape of the base so I'll read what they say first and then I'll say things after that the two bases of prison are congruent polygons and parallel planes for example the bases of pentagonal prism are pentagons so you can see on top and on bottom we have pentagons those are the unique shapes of these prisms therefore it is pentagonal prism now I'm going to say more about bases with that in general the base of pyramid is polygon for example the base of triangular pyramid is triangle so this is the base here as well now some things they do so they did mention some things or some things they did not mention here and there's also phrasing did not see which kind of want to get into but I'm sure they're going to talk about it more with the surface area number one the bases do not have to be on the top and bottom you could have and don't have anything with me right now but you could have something that it flipped over you know flipped on its side if this Pentagon facing was on the front and this was on the back it's just flipped over version that doesn't change the fact it's not pentagonal prism so how do you identify the bases well if it's rectangular prism anything could technically be called its BAS at that point you might as well call the top and bottom the base but for prism the lateral faces these ones on the sides I'm circling around the the side ones right here those would have to be rectangles in some cases parallelograms but you'll see when those appear when they are forget what the name is oblique prisms or something like that but otherwise they're rectangular in nature you would not have pentagon in this case because it wouldn't make parallel bases they all that kind of stuff so these are the unique ones on top and bottom now the triangular pyramid everything's triangle so it's hard to say but the base would be the unique one outside of it and then all triangles meet to this Apex here afterward so I'm sure they're going to say more about them later so to name some of these ones name the polyhedrin in this case would call this one rectangular prism they're saying classify whether it's polyhedrin or not so it is polyhedrin this is polyhedron this is not polyhedron as this is polyhedrin it is rectangular prism it has all rectangular faces including what would be guess the top and bottom is bases so we'll call it that that's what they call it this one here is also polyhedrin and it is pyramid but what kind this is the unique bottom and base and most pyramids will be facing upward like that some will be upside down guess they could be any direction but that's hexag hexagon hexagonal shape so it's hexagonal pyramid and that's what they say here last one circular pyramid if you will and that's something I'll probably be mentioning later especially when we talk about probably more surface area but you know volume actually still applies it's cone it's cone it's an inverted cone but it's also cone it has curved surface so it's not polyhedron okay this is square pyramid you can see all sides congruent and the right angle this is well semicylinder it's cylinder with semicircular base there so therefore it's semicylinder suppose not polyhedron okay and then this so this is an example this is an example of this is not the base this is not the base just because it's rectangle or just because it's the bottom doesn't make it the base because now what are you calling this it's not pyramid it doesn't meet at one common point at the top here the bases are actually here this triangle and this triangle they are separated by rectangles rectangle here rectangle in the back and rectangle in the bottom they're separated by rectangles therefore these triangles the unique ones are the bases this is triangular prism it's triang angular prism it's just knocked over it's like block of cheese that he just topped over there so that's the important one to name there okay finding volumes of prisms and cylinders the volume of solid is the number of cubic units contained in its interior so it's like how much you can fill in something if you're filling something up with liquid or sand grain something like that how much fill there is in cubic units in that way yeah volume measured in cubic units such as cubic centimeters da da instead of square feet like area we talk about how much you can cover with something wrapping paper with paint you know something like that those would be square units these are cubic units here's this part Cavalier's Cavalier's principal I'll come cavaler Cavalier's principal named after bonaventura Cavalier States if two solids have the same height and the same cross-sectional area at every level then they have the same volume so you can see here are two examples of prisms and wish that they named what kind this was think it's called an oblique prism not quite certain but you can see here this may be rectangular face if you can understand the 3D of that whereas this is parallelogram otherwise they'll generally be rectangles but as long as they have the same base area and the same height as we get into volume and talk about that the the crosssection being the same because this cross-section being the same square base area with that same height then they will occupy the same amount of space they have the same volume the way would equate this and get didn't prepare this is say you had stack of paper stack of paper and you took that paper you know you wedged and held the sides and you bent them over doesn't change how much paper you are occupying as an overall just cuz it's going that way excuse me this is stack of paper just going that way instead of going straight up still equal in volume cavaler must have had lot of paper in his time to to demonstrate that all right volume of prism here's the formula and it's very basic formula if you know the volume of rectangle is base times height volume of prism is also base times height the difference is this being capitalized you know lowercase letters are generally with respect they generally have to do with one-dimensional units lengths so height height is length this here is two-dimensional unit in this case is three dimensions 1 time not 1 * 2 is three onedimensional unit time two dimensional unit three dimensional unit but because it's two- dimensional it's capital how come it's not just base length it's area of Base so whereas rectangle is base times height as far as length goes volume is area of base time height so it's very simple formula and if you equate it to the way that rectangles and parallelograms because parallelograms have the same thing it's also base times height base time height that's why both of these ones work the way that you see them whether it's prism even the square prism with rectangular faces or square prism oblique prism with some parallelograms involved it's still area of Base times height the question at hand is how do you find the area of its base and the answer is it depends on what kind of Base it is and that's where things change that's why 113 section 113 was so important that we knew how to find the area of regular pentagon of regular octagon of kite of rhombus on top of triangles rectangles and squares so and trapezoids in fact here's an example of trapezoid so we have to know how to find areas of bases and that's why naming something by its base is so important because it's part of our formula so example finding volumes of prisms we can see triangular prism here there's right angle so it's right triangle that can help us find the area of that base there's trapezoidal prism here so we're going to do area of Base times height I'm generally going to separate my work very organizationally where you can tell where I'm trying to find capital area of base and identify what the height is the height by the way haven't really mentioned it's the distance between your bases it doesn't always go up and down the height in this problem would have been this right here it's the distance between your bases and this also would have been the height and in the pyramid we'll talk more about those later but let's find the area of this base here in right triangle now there're two different heights right there's the height of the prism and the height of your base 1/2 base time height in the area of the triangle so 1 12 3 * 4 is 6 6qu cm that's the area of your base the height of the prism is 2 cm so area of Base 6 * height 2 is 12 that's 12 cubic cm of volume Part area of trapezoid is you average your bases and multiply by the height so you have base length here of 14 another one of six Bas length meaning in your base that that's the problem saying base and multiple ways and height in multiple ways so you average 14 and six and you get 10 and then you multiply by the height of the base which is three that's the perpendicular that's the distance between your bases and then you get 30 square cm as the area of that base the height of the prism is five that's congruent that's consistent throughout so area of Base 30 * 5 is 150 that's 150 cubic cm volume okay let's talk about were we give me one second forget how introduce this no wonder why they're not talking about pyramids really I'm sorry if in the very beginning of this section thing said volume of prisms and pyramids hope said volume of prisms in cylinders don't know if did anymore cuzz my mind started getting on pyramids when they started talking about prisms excuse me when they started talking about polyhedrons okay cylinder consider cylinder with height and base radius in rectangular prism with the same height has square base with sides of length * < TK of Pi okay there okay they're stretching little bit cylinder and Prisma have the same cross-sectional area pi squ at every level and same height height by Cavalier's principle the prism and cylinder have the same volume so here's the thing to me even though cylinder is not prism coin it as circular prism if you name it by its base the base is circle and it has the same prism properties that will really help when we talk about surface area when REM mentioned circular prism the other way it also works is for volume it's still area of Base times height it just is very specific base there's only one type PK 2 so area base time height is PK * height so if you see me write that specifically for cylind we should be all good but if you just need to know area base times height you can still work that out area of the circle is PK squ you can have an oblique cylinder like that one as well and it still works but the height is the perpendicular distance so let's find the volume of each cylinder for this volume we're going to do pi squar for the area of the base Pi * 9 is 81 Pi they rounded it no they didn't yet but 81 pi times the height of 6 gives you 486 Pi cubic feet rounded looks like they're using two decimal places so let's get ready to do the same when we work on it this one as well excuse me the height is seven that is the distance between your two congruent faces Pi * 4 squ is 16 Pi * 7 is 112 pi rounded to 351.5 cubic cm so I'll let them find volumes of solids for themselves later then we'll talk about similar solids two solids of the same type with equal ratios of corresponding linear measures such as height of such as height or radi are called similar so solid so they have to grow in proportion with each other in every which dimension the ratio of the corresponding linear measures of two similar solids is called scale factor sometimes call it similarity ratio so they both kind of mean the same thing if they have scale factor of and they're similar then the ratio of volumes is cubed we've gotten heavy I've mentioned lately think especially in section 112 maybe 113 that the ratio of areas between two objects would ideally be squ that if you double length of some something radius or side or whatever into objects that are similar then their areas excuse me the ratio of areas you you square that it's it's quadruple that amount well the volume would be octuple whatever you multiply by 8 2 cubed is8 so think we're going to see that in this instance here cylinder and are similar find the volume of cylinder notice that they don't give you enough information to say well what's the height how do find its volume you could do certain things you could use this volume and that radius to find this height and because this thing doubled in length here then this would double length there that's little Overkill what you could also do is just compare the six to the three say that it doubled and if the length doubled then the volume got 8 times bigger 2 cubed which is 8 so you have you know like 8:1 ratio as this is also blank to 45 pi and then you can find that volume as 360 pi they didn't seem to round that number guess because this was already exact so can follow along with it okay finding the volume of composite solid so here's one of the last bits composite solid is basically an irregularly shaped one either you're adding or subtracting volumes with each other depending on the type of shape that it is you can see concrete block here that's basically rectangular prism with rectangular prisms cut out from them they don't show you the cross-sectional parts that are cut out but you know it would be like this idea here when this guy is cut out from inside and then this guy let me use my tablet and then this guy also cut out from inside right there we got to remove these two volumes they're probably congruent but we got to remove these two volumes from the volume of the other guy the area of the area area base okay they're getting little whatever I'm going to do volume separately but the volume of the large rectangular prism minus 2 times the volume of the two small rectangular prisms you can do it that way they seem to multiply by height after the fact which is fine excuse me about4 cuic feet and this one you got to sorry this one I'm hiding right there you got to cut the volume of that square prism out from outside from this triangular prism and it is triangular prism this is base and that's base right there separated by the rectangles so they're not talking about every Target word thing right here it looks like pyramids are saved for different thing again apologies if said that from the jump should be more careful with that okay so here we go vocabulary and core concept check in what type of units is the volume of solid measured they are cubic units cubic units so you're going to see where it says like feet cubed meters cubed centimeters cubed cubic feet cubic meters cubic centimeters things like that will happen number two which solid does not belong with the other three explain your reasoning we have few pyramids here which are all poly pedrons so if those are all A's to us with square base triangular base and pentagonal base this one is circular pyramid otherwise known as cone that one's going to be different it's not polyhedron not polyhedron it has curved surface or surface with curved edges at the very least okay guys we are going to jump into what looks like 3 through six next down below but before we do hold on just activated magnifier on my screen don't want to do that before we do want to remind you to check out time stamps for the kind of problems you're trying to find and in the description section down below and also have calculator ready imagine that since we came off of sorry imagine that since we came off of section 113 with areas of regular polygons we might have to do some of the trig and special right triangles with the 1 half equals 1 half APM tense perimeter I'm sure it'll pop up somewhere so guys let's go ahead let's get started here numbers 3 through six match the polyhedron with its name they have some names down below so for number three what you're looking at there you're looking at pyramid excuse me it's polyhedron it's pyramid these sides are congruent these sides are congruent that's right angle it's rectangle at most known so it's probably rectangular pyramid so that's going to be so rectangular pyramid let me say at the Forefront there just want people to write the letters you here all right number four these parts will go pretty fast number four the bases on top this polyhedron they're all polyhedrons the bases on top and bottom are pentagons this is prism congruent two congruent bases so it should be pentagonal prism probably good time to type these out but I'm not number five the base this is not pyramid number six is pyramid this one's not this is prism this is prism the bases are in front and back those bases are triangles they are the unique types whereas the rectangles are the lateral faces this is triangular prism that is triangular prism have feeling I'm going to have couple ywn here haven't slept much number six that is pyramid well by process of elimination it's going to be but how come because it is pyramid all triangle faces meet at the top there and the bottom is hexagon so it is hexagonal pyramid this is okay let's keep moving forward let's look at numbers seven through 10 tell whether the solid is polyhedrin if it is named the polyhedrin so just one step more I'll name every shape for you whether or not it is polyhedron so on number seven we have another pyramid it has one two 3 4 five bases this will be number one it is polyhedron and it is pentagonal pyramid the base it's name based off its base all right number eight also polyhedron now it doesn't look like regular polygon on bottom Maybe is can't tell but looks like hexagon six sides so this is hexagonal it is prism it's polyhedron and hexagonal prism number nine number nine is not polyhedrin you have couple things involved here and last time they called this so maybe can't really give this an official name for you other than mailbox or whatever I'd call this it's composite solid composed of perhaps cube or rectangular prism with semicylinder on top of it so not polyhedron so guess can't give you an official name because it's composite solid that kind of just has the appearance of don't know is that what mailboxes look like I'm not trying to remember suddenly so not much can say there number 10 is polyhedrin however however these are not the bases on bottom and top even though they're possibly parallel with each other because it kind of looks like trapezoid thing and I'll talk about that even though they're possibly parallel with each other they are not congruent they're not congruent the prism needs to have congruent bases such as this and this so the congruent bases are on front and back the ones that are irregular that are trapezoids these trapezoids are separated by this distance here which you would call height this is trapezoidal prism it is polyhedron and assuming the lines are parallel not just anyal quadrilateral prism is trapezoidal prism so it is super important that you know the name of well the name of it for couple reasons number one to know what kind of area you're doing and number two to not where your bases are because if you don't know where your bases are you can't find the area of that base so it'll be super important and we're going to find volumes right now so number 11 on number 11 with that drawing we are going to I'll probably get these ones copied and pasted over for the most part just so we could identify certain things where we're going to find the volume of each prism we need to answer them in cubic units as well so on this number 11 identify where the unique bases are they are the front and back not the top and bottom once again this is another trapezoidal prism it is right trapezoid given that that's the right angle so to find the area so volume is area of Base times height it's probably the only time that I'm going to officially just write that itself the area of the base is going to be You Average the bases of your trapezoid in this case the ones that are parallel to each other this 1.2 and the 2.3 are going to be averaged and you multiply that by the height of your trapezoid that'll be 1.8 the distance between your bases of the trapezoid of the base it's weird because there are different things I'm calling bases there's the base face and there are the two bases of your base the trapezoid there's height of your base that's the trapezoid and there's the height of the prism which is this two which I'm going to talk about next so base is half of 3.5 which is 1.75 and 1.75 * 1.8 that's so it's probably close to 3.2 let's see 1.75 * 1.8 3.15 so 3.15 square cm is the area of your base the height of your prism is 2 cm that's the distance between your two bases is in your prism probably feel like with the inner work I'd want to generally color code that stuff but also wouldn't have that formula part on Top generally either so don't know it's kind of Hit and Miss I'll just leave it so the overall volume is area of Base times height of any prism so volume is area base times height which is 6.3 cubic cm sorry 6.3 cubic cm that's number 11 number 12 12 seems little bit easier now there's more than one way to do something like 12 in that because it's rectangular prism which ones are the bases as long as they're opposite each other they're bases you can have the front and back you can have the bottom and top which is more conventional I'll probably do that or you can do the left and right right these ones as long as the height is what separates your bases it won't matter which one you do some people say isn't the volume length time width time height yes because the length time width would be the area of the base times the height but don't try and be specific and cute with them if you know them as area base times height each time think you're going to be well off I'm not going to copy paste this one the area the base I'm going to use the four and the two the bottom so 4 * 2 is 8 that's 8 square met the height of this one will be 1.5 distancing themselves from each other so the volume is going to be this area base time height 8 * 1.5 is 12 you get 12 cubic meters feel like they should have started off with that one just it's more basic shape more basic way to identify it more basic calculation instead they started with the other one number 13 this is one of those think I'm going to keep saying think until do figure it out think it's an oblique polyhedrin in this case an oblique prism is the name that just because it's slanted by Cavalier's principle we should also note that the volume would still be the same as long as you have the height as the perpendicular distance between your bases so the area of your base this is triangular prism so the area of the base is 1/2 base time height by base mean base length of your base base Edge so 12 of 7 * 10 can apply here because those bases and Heights are perpendicular to each other as well half of 70 is 35 so 35 square in for the area of the base and the height is 5 in that's still the distance between your two bases here so the volume total volume is going to be area of Base times height and area base times height here is 175 cubic in and then number 14 same stuff as before you do have some congruent bases on top top and bottom let's use those whereas the height they're declaring should ideally be the 14 they are rectangular bases so it is rectangular oblique prism so the area of the base in this case is 6 * 11 which is 66 square and your height is so the volume in this case is area base times height in all these cases will be area base times height which is 66 * 924 cubic and that's your first for crack polyhedrons numbers 15 through 18 find the volume of the cylinder they're all cylinders volume of cylinder is also area of Base times height the difference is area of Base in this case is always PK so your radius and your height are the two things we need to know this time to get area base the radius is 3T the height is 10.2 ft on this first question number 15 so the area of Base given the radius is piun * 3^ 2 or 9 pi Square ft we are going to do area of Base * height in this case so 9 pi * 10.2 there's an exact answer somewhere within there but they've been doing lot of rounding to the nearest 100 so I'm going to do the same thing so 9 pi Pi * 10.2 is going to give me 288 398 so I'm going to round that up to 28840 cubic feet volume of cylinder on number 15 number 16 we have cylinder but this time with diameter given not the radius so the radius will be half that diameter 26.8 over2 which is 13.4 CM the height is 9.8 CM still have to find area Base by the way keep skipping that area of Base pi 13.4 13.4 squared is 17956 don't forget the pi portion of this still this is square cm so area of Base boom height boom let's go ahead and round that guy when we get volume volume is this times this 9.8 so let's do that that number times pi * 9.8 big number I'm to make sure got all that okay 55 28.22 8.22 Cub CM 22 number 17 and 18 oblique cylinders doesn't change what we do we have area of Base times height once again so let's just jump straight into that volume in this case is area of Base r^ 2 * height so this is this is so this one 25 * 8 is 200 so 200 Pi is an exact value obviously we're going to be you know rounding this stuff so it should be 628 31 think that's because 3.14 doubled 314 doubled things like that so 200 Pi is 628 318 so32 62832 cubic feet and then number 18 you do have to first identify your radius by dividing 12 by two now notice they show you that 18 okay I'm going to copy and paste this one actually notice they show you the 18 as well but 18 is not the height 18 is form of not maybe the official term but form of slant height but it's not the height of your prism it's not the distance between not the perpendicular distance between your two bases your circles that's why they're giving you that 60° angle measure as well though they need you to come up with what this actual height over here is if this is 18 by the way that's 18 so we'll get into that in moment let's start with the radius radius is half of 12 which is 6 now the height is you know this is 30 60 90 triangle this is the hypotenuse of it as 18 half of that is opposite the 30° one so this is 9 we don't really need that for our calculations but we do need it for the height so the height therefore is going to be 9 < tk3 that's an exact value in that 30 60 90 setup 1 to 2 to < tk3 so height is 9 < tk3 so as far as PK r^ 2 * height that's < * 6^ 2 time height as an exact answer 30 6 * 9 is 324 think so 324 < tk3 * pi that's cubic met but let's check that out so time < TK 3 and time Pi 1763 01 Cub is the volume yes we needed this height that distance with that radius okay 19 and 20 make sketch of the solid and find its volume round your answer to the nearest 100 so number 19 we are going to sketch prism that has height of 11.2 cm and an equilateral triangle for base can do that part where each base Edge is 8 cm so so what would do is draw the top portion if you want to call these bases and like to kind of slant them off little bit otherwise you kind of get something where you can't really see the 3D aspect of it but I'll get the slant right here and it is equilateral know it's equilateral because marked everything congruent you see you're just seeing it kind of from rotated angle there but we also have this downward height like this right here this is good practice for you to do and then dotted line on the back end of this because it's kind of where it goes and on the bottom you need that same appearance of what you have on top so it's triangular prism everything's congruent all of these lengths are cm the height is 11.2 cm and you need to find the volume okay so what you might want to do here is kind of just redraw that equilateral triangle in flat way this is where we started to have to do some of this other kind of work in an equilateral triangle where all of these are eight this is one of those you could do the 1/2 APM times perimeter if you want to find the central angle I'm more in tune with equilateral triangles of dividing this here working off the 30 60 90 so this is 4 and this is 4 < tk3 inside that's your height so your area of your triangle the area of the base is 1 12 base 8 * height of 4 < tk3 which is 16 < tk3 Square cenm that's the area of the base with your height being 11.2 the overall volume is going to be area of Base times height so I'll jump straight into the rounding of that especially with the root three involved okay I'm getting 310.10 38 cubic cm so finding the area that base again it depends on how you want to go about it you could do one and half APM time perimeter then you need the apem kind of found the area of this entire triangle by splitting it up in the 30 60 90 way given that it was equilateral either works number 20 pentagonal prism so this time it seems like we're going to do use the 1 half AP think pentagonal prism has height of 9 ft and each base Edge is 3 feet so there there it goes and telling you it's regular pentagon so one 2 3 four 5 again kind of make it off to the side if you want to get something more unique with it I'll slant that in little bit find each Edge they do want you to draw it so you can do it you know kind of like that whoopsie okay now everything's congruent in the pentagonal prism as far as the it's regular pentagon all congruent each of those lengths are 3T and then you have height of the prism being 9 ft height of the prism is 9 ft so as the pent as the Pentagon redrawn over here this is where would get into we have to find the area of that base so need to find the apam given that know let's see don't need radius but need central angle given that know that an entire length here is not eight it's three so half of three is 1.5 fet just remember this fet need the apem here do need to find that perimeter as well it's pretty easy calculation but just remind myself of that so need the central angle divided two central angle is 360 over 5 which is 72 half of that is going to be 36° there it's bisected this was from section 113 so to find the aatham we do tangent we take tangent of 36 and we get 1.5 over so is 1.5 over tangent of 36° the perimeter is 5 * 3 which is 15 these are all in feet so the area of our base the base itself is 1 12 apam time perimeter 12 of 1.5 over tangent of 36 * 15 don't really want to round that yet CU we're going to multiply by height guess at this point might as well just say let's multiply this all by height now so the volume is area of Base so this guy here is area of Base times your height and the height is nine so nine is height like that so if we do this all in one Fell Swoop will be good and hope you're okay with that just because of how it work that out remember area base 1 half APM time perimeter for the Pentagon and then the height going there so let's go and go to the calculator and type all that stuff in will be typing in um2 so 0.5 time 1.5 over tangent of 36 time 15 * 9 just kind of writing it the way that had it written there believe I'm in degre reason have been this whole time let's double check yep so this is 139.3050 cubed all right so I'm writing that down so you know once again this all stems back to what we did in section 11.3 and understanding regular pentagons and how those volumes work and this one follows that thing so one half APM times perimeter needed the APM using some form of trig and we got it all right let's go and move to numbers 21 and 22 on error analysis describe and correct the error in identifying the solid this is good okay cool number 21 rectangular pyramid incorrect so this polyhedron it is polyhedron but this polyhedron is not pyramid pyramid would only have one unique base with the other faces as triangles meeting at an apex this has two congruent bases that are triangles in front and back and rectangular lateral faces separating them as they're parallel this is triangular prism yes it is very common for people to mistake that for pyramid just because you have triangles only two of them doesn't make it it's the way that it's drawn though right it's because it's toppled over if it was drawn the way that we drew this one right here you would call that prism you wouldn't call this pyramid so this this is what that is it's just faced differently right what if took this shape this object right what if took this and Drew it this way is that suddenly not prism you know don't turn it over you go my gosh now it's pyramid no it's still the same shape as it always was so just kind of you know keep that in mind all right number 22 by the way rectangular pyramid described it but rectangular pyramid would have had rectangular base and then the triangles all meeting up at say some top Point like like that so this would have been an example of rectangular pyramid like that okay that's rectangular py okay number 22 error analysis describe and correct the error in finding the volume of the cylinder so they used different formula here they said 2 pi * the height 2 pi is the circumference of circle which we will be using this for surface area stuff make no mistake it will be used but that's circumference not area so they applied I'm sure all their math after that is correct but they applied the circumference of of their circular base in the volume formula when it should have been area of Base so you know th this goes back to if you don't have the formulas at hand if you don't have them in front of you such such if you just know that all prisms and cylinders and cylinders are circular prisms volume is just area base times height you're really good to go to begin with like then and there because then you can say Okay area base is pi squ so Pi * 4^ 2 and then height you know still three so now you have that so 16 Pi instead of the 8 Pi that they have there times the three gives us instead Pi cubic feet there's the final without rounding cuz they didn't all right 23 to 28 find the missing dimension of the prism or cylinder we're still using the volume formulas from before but this time they give us the volume to find something that's missing so in number 23 got to solve for you now in this case volume this this is rectangular prism so we're still going to do area of base time height right area base time height now the area of the base is going to be that 7 * 8 we can work that out area base is 7 * 8 which is 56 Square ft but the volume is also established the volume is 560 cubic feet that's the first thing they give you so we have to find the height it says but yeah so yeah so equals how about that so volume is going to be excuse me volume 560 is going to be equal to do 56 * uide 56 you get = 10 this is the height this is length this is in feet is 10 feet you are 10 feet number 24 once again same kind of thing rectangular prism you do have area of the base so area let's start with volume because that's written first volume is 2 700 cubic yard the area of the base is length times width base time height however you want to call that that's 180 square yards the height is this unknown that we're going to call when volume is area base time height we can say 2700 = 180 * and then you divide by 180 you get equals is that 15 think that's 15 15 yards so being the height I'm going to double check that 2700 divided 180 it should be 15 believe yes so 15 yards the height the okay number 25 this is as we just saw from before this is not rectangular pyramid it's triangular prism because it's name based off the base we got to find the area of that base or you know use the area of the base and find is also height in fact all these are Heights all these ones even number 26 but for this one number 25 the volume is 80 cubic cm this is triangular prism so when we find the area of the base we'll be doing 1 12 base time height which is 20 square cm the height of the prism we're trying to find is this unknown so volume equals area of base time height so height is 4 cmid 20 say height is in number 26 is again this isn't like the bases bottom and top the bases front and back this is hexagon regular hexagon it's hexagonal prism we have to find the area of this hexagonal prism and had formula remember on 113 if you watch probably didn't that far farther into the video we kind of had generic way of coming up with the formula think it was 6x^2 < tk3 it was something weird we'll have to just do it on our own you probably can't really apply that generally in other things by by remembering but yeah over here we're going to go ahead and find the area of that that base of the hexagon in second but the volume is 72.6 cubic in the area of the base is going to be 1 12 apam time perimeter apam we have to find perimeter would be 2 * 6 which is 12 in so 12 so what is the aatham of this will be easy the 2 in is divided in half to get one and this is 60° central angle from hexagon 360 over 6 / 2 is 30 this is 30 6090 triangle which has 1 to2 to root3 ratio relationship is literally < tk3 it's 1 * < tk3 so sare < TK of 3 goes here so area of base is 6 < tk3 square inch your height of your prism is the distance separated by your two hexagons that is the unknown so volume equals area of base time height so = 72.6 over 6 < tk3 unfortunately it will be irrational but I'm sure there is rounding we're going to round to the nearest 100th I'm sure the rounding is going to be intended to be something close to whole number given the 66 whatever so we'll see let's jump to the calculator and see what the heck is we got going we have 72.6 divided by I'm going to use parenthesis for 63 you can't forget that 6 root3 means 6 * root3 and if you write it otherwise without the parentheses it will not register in the denominator with you so 6991 okay if we're rounding to the nearest 100th guess got to put 6.99 but clearly it wanted to say around seven Ines but I'm going to write 6.99 just because of the way that the rounding went at least to the nearest 100 darn wanted that seven number 27 we have 3,000 cubic feet of volume of cylinder the radius is 9.3 so the area of the base is < * 9.3 which is not rounded just got to do 9.3 squ here which is 86.4 9 was that the other number before 86.4 don't think so 86.4 Pi Square ft and the height of the cylinder is the distance between your circles which is which is the unknown so volume equals area of base time height 3,000 = 86.4 piun * and this this is going to be the same thing as the previous problem if you use the calculator for this straight up and didn't round early and hope you didn't but didn't even calculate early make sure Pi is in parentheses with this with this set so over here I'm going to do 3,000 divided by parentheses 86.4 pi and now we're going to get the better answer 10.98 10.98 ft is that value number 28 volume we haven't done composite solids yet but we also haven't done word problems we're getting pretty deep into how how many problems are there 44 we've we probably not as many proofs in like critical thinking and stuff I'll probably use the bathroom after this one if it's okay doesn't matter to you I'm just going to pause volume is cubic meters the radius is the unknown this time radius is which is interesting because the area of the base is pi * 2ar so Pi * z^ 2ar this is the first time that height isn't the one that's not established height is established this time it's 15 so with that in mind the volume 16 96.5 equals area of Base Pi ^ 2 * the height 15 I'll put the 15 actually in front let me get rid of that little dot going put the 15 out in front of that Pi * so we have two steps to do we have to divide both sides by 15 pi and then take the square root of both sides so is the square root of this and that's approximately what say same thing as before when type in 15 Pi into here have to make sure it's in its own set of parentheses so Square Ro of 16 96.5 / parentheses 15 pi and now get six right that that's the clean kind of number I'm looking for that's radius length that is in one-dimensional units that is meters so 6.00 radius okay let me pause be right back okay so we finished 28 let's move on numbers 29 and 30 the solids are similar find the volume of solid so we don't actually need to find the solid through calculations of the are you know all that kind of stuff it's pentagon and no we don't have to do all that all we have to do is compare the ratios of corresponding parts in this case and don't know how want to specifically write it but the it's called the ratio of volumes this is what tend to do ratio excuse me the the similarity ratio the similarity ratio Subs know they use don't know I'll just say similarity ratio this it's it's my own habit I'm going to compare those two heights notice that the prism is N9 cm and then prism is 3 cm so the similarity ratio is 3 which is 3: 1 let me write 3 over 1 now the volume ratio is going to cube 3 over 1 which is 27 over one in proportionality that means prism A's volume is 27 times more than prism so just have to divide 2673 divided you know by 27 so yeah whatever 27 over 1 is 2673 over volume of prism is so listen there's so many ways to write what you do the bottom line is we're going to be you know cross multiply divide all that kind of stuff volume of prism is 2673 over 27 no one way to write this stuff out outside of the fact that you have to cube the ratio the similarity ratio the scale factor of those ones and that number believe that's 99 just because it's one away from what 2700 would be so 2673 / 27 is 99 yeah so that's 99 cubic cm that's the volume of that one lot of this you could kind of do like could have probably done that within like 5 seconds in my head in that same kind of way see if can do that for number 30 don't know if can maybe can't for this one definitely can't with the numbers but yeah you know what can't for this one it's it's not as easy number 30 we have two cylinders once again don't need formulas for this stuff outside of how these things compare so the similarity ratio is going to be how about this let me just say volume ratio straight up volume ratio is going to take your similarity ratio from one to the other like 12 to 15 and we're going to cube that so and that reduces by the way to four fifths and it's probably best that you do Cube you do reduce that or you could even call it 08 but 45s cubed is 64 over 125 now8 cubed would be 0.524 so 64 over 25 apparently is 0.524 so cylinder is 52.4% that of cylinder all that kind of stuff if you want to say it that way so this ratio let's set it in proportion to volume of cylinder 4,608 pi cubic in over volume of cylinder so will multiply this up here don't know how much you need to see algebraically the rest is algebra so multiply this by that and then multiply both sides by the reciprocal like that I'm expecting clean number here I'm going to keep Pi out of my calculation when it comes to doing this because want to do it in terms of pi as long as it's clean number so 4608 * 125 / 64 is an integer in fact it's very nice integer this is 9,000 so 9,000 Pi never multiplied by pi in my calculator so 9,000 Pi cubic in is the volume of cylinder all right 31 to 34 are our composite solids let's find their volumes so in order to do this we have to add or subtract certain things based on what you're looking at so sometimes there's kind of bit of setup here what tend to do in situations like this and guess I'll copy and paste probably all of these over is I'll call them object one and object two you know based on what you're doing with it so you get good understanding of what we're doing as general to get the volume of the total so I'll coin this one object one and this one object two so the volume of the total just so you're aware is we're going to be adding these two volumes up the volume of object one plus the volume of object two they are both rectangular prisms let's start with volume of object one by establishing what we have here we have base if call the bottom stuff the base so you have base edge of five and of three in this rectangular prism so the base of volume one okay the base of volume 1 is going to be 5 * 3 the area of the base should say is 15 Square ft and the height of volume 1 is 2T it's very easy to get get disorganized in this unless you do something like what I'm doing here so the volume of item one object one is area of Base times height which is 30 cubic feet let's leave that for moment it's not the final answer but leave that there item two item two has in fact it's double everything could tell you this answer now is going to be think it's going to be 150 cubic feet cuz think everything's doubled the Height's not doubled never mind never mind never mind never mind okay base area of two is Boom CH boom 10 * 6 which is 60 square ft the height of of rectangular prism number two is also 2 feet so the volume of object two is area of Base times height which is 120 it still is wait minute and must have done something wrong in my head with all that in mind mean there's nothing wrong that did calculating wise right 5 * doubled 10 * 6 doubled that's weird that still 150 must have done something wrong in my head that I'm not wouldn't have thought 150 would have thought 200 270 so even made mistake on that anyway the volume of the total is going to be 30 + 120 which is 150 maybe if speak less and trying to sound smart in like trying to whizbang some stuff maybe can actually get some stuff right now here's the thing though did mention organization and must forewarn you there's chance can get something wrong one slight miscalculation one slight misstep in number can screw the entire thing up so I'm only hoping that this doesn't happen here in older videos I'm sure was rough around the edges and made some mistakes that's really why laying this out is super duper important you know one small mistep can absolutely do that so hopefully you're following my layout and my plan so that if you get different answer even if yours being correct you can tell me about it in the comments stuff and you know we can you can still get it right so that you're not just copying everything say that being said let's hope get everything right so number 32 now I've gotten so let's let's call this one one and this one two cuz number one's going to be more basic I've gotten so heavy on telling you guys and preaching the fact that you know this is square prism and stuff like that but if you know different kind of formula that will apply more this is cube this is cube and yes it's area base times height but if you want to get even little faster with this one if you understand the volume of cube is by cubing all your stuff there and you get 64 cubic inches mean I'm good with something that you know how to do I'm just stating don't get too specific on other formulas if you don't know how to get back into the basics of them or that you mistake them but cube area of Base 4 * 4 16 * height 4 we do get 64 cubic in the volume of the second one's little tricky because it is area of Base times height if it we whole cylinder but we have to divide it by two so in this case here you know the cylinder would have been this whole thing like that right and it would have gone down and there and all that but there's only half of it so we have to take the area of the cylinder divided by two and remember it's all about the circular base that means these are the bases the height is the distance between them so for volume two we need to get the area of the base as diameter is four because that's four therefore the radius is two let me let me go one step down here the radius of object two is 2 in so that base of the cylinder semicylinder is < * 2^ 2 which is 4 Pi Square in and the height the separation between those two semicircles the height of this is this distance here that is 4 in so the air the volume of your second object is area of Base 4 Pi * height of 4 all divided 2 16 over 2 is 8 you're left with 8 piun cubic in so we have 64 cubic in and 8 Pi cubic in the volume of the total should be the sum of those two 64 + 8 Pi cubic in when rounded probably probably somewhere around 90 so so 64 + 8 piun 8913 cubic in okay let's go to number 33 We have basically toilet paper roll there's the layer on the the the rim that you know it's stack of discs there's hollow inside we have to remove the volume of the inside so it's subtraction in this case the volume of the total we're going to do some sort of sub one minus sub2 if identify this as my guess got to copy and paste this and call what I'm going to call one and two but you have to be good at looking into whoops looking into the shape itself to really make sense of that so when say this I'm talking about here's object one and the inside is object two perhaps if make it little more transparent looking like that you can dig all right so the volume of the first now they both share the same height don't know if that matters for us for for you the radius of volume one just imagine that that inside Hollow part's not there the radius of volume one of object one is 8 in the area of that base is < * 8 squ which is 64 Pi Square in and the height of everything of of both of these objects is 11 in I'm going to use that for both items so the volume of object one and what the what the what the examples did in the lecture portion is they just found the areas of both of them and subtracted the areas of the bases first and then multiply by the height after you may do that I'd say maybe try not get too tricky with that just would just be careful so area of Base times height gives you Pi cubic in there's the volume of the first guy the second guy we have radius of three in there for base area of 3 pi 2 9 pi Square in we know that its height is 11 in so the area EXC me the volume of object 2 is area of base time height which is 99 Pi cubic in so the volume of the total we're going to take 704 pi and subtract 99 PI from there so volume of total is 704 piun - 9 9 pi those can clean up because they both share Pi in it so they can combine that 605 Pi cubic in as an exact value when we go ahead and round that when we go and round that times pi we get, 1900. 66 1 190 .66 cubic in as whole so removing the roll out of the toilet paper having hollow in inside having hollow interior and guess it's up to you it's it's up to the the book writers problem makers to make sure that you understand what that looks like but that's exactly what we're talking about for that composite solid number 34 once again something is removed here the tricky part of seeing all these dotted lines this is the part that's removed here this dotted line part over here is just part of the backdrop of that oblique rectangular prism that we're looking at but they're both oblique rectangular prisms each with the same height of five it's still volume 1 minus volume two aspect to this thing let's coin them objects one and two like we did last time this 34 part is going to get part of the cut and paste job I'll try and hide that in moment okay give me second good is new close enough okay so we have object one and inside object two both of them share height of 5 feet that's not going to change that's the perpendicular distance between your bases whether it's the inside or the outside now the volume 1 minus volume two thing will apply so for volume one let's find the area of the base and then we'll multiply it by the height area of Base it's rectangular just by Nature guess we assume it is so 4 * 2 which is 8 square ft so the volume of object one is area of base time height which is 40 cubic feet area of base two that's square that's square prism they're showing both of these are ones so one squared so one square foot so the volume of object two is area of base time height which is five so we get five cubic feet there the volume of the total we are going to be taking 40 cubic feet and subtracting 5 cubic feet we get exactly 35 cubic feet assuming no mistakes were made we should be just good on that one for number 34 okay all right let's move down to modeling with mathematics think it's our first word problem thing how many problems are we doing in total 44 so we have 10 more to go saving all the word problems and critical thinking and how do you see it and your friend argues this stuff all probably for the rest of the way modeling with mathematics the Great Blue Hole is cylindrical Trench located off the coast of Biz it is approximately 1,000 ft wide and for 400 ft deep about how many gallons of water does the Great Blue Hole contain 1 cubic foot is equivalent to 7.48 Gall we're going to save that last part till the very end whatever we get in cubic feet let's multiply it by 7.48 about Seas so the 1,000 ft wide mean what they're talking about here is it looks cylindrical so the 1,000 ft assume is going to be some form of diameter so 1,000 ft which means half of that is 500 ft if you're okay with me just dividing now we can go go and talk about that now it goes deep as well goes 400 ft deep probably not drawn scale there if that's if that's 500 then 400's probably looking just like that it's kind of smaller cylinder if you will know it's supposed to be blue but whatever so 400 400 feet we're fing this guy's volume so the radius let me do that calculation radius is 1,00 over two which is 500 and the height is 400 that's the depth depth is the height so volume is area of base area of Base times height think that's 10 not 10,000 it's like 10 million 10 gez what is it it's like 1 million let's go to that five I'm going to do the exact before do the any ing cuz I'm also going to multiply by 7.48 so 500 2 * 400 1 2 3 4 5 6 7 8 100 million so the volume is 100 million Pi it's weird to say 100 million Pi cubic feet of WWA I'm keeping the pie in there for moment how many gallons so when you multiply by 7.48 that'll be 748 million gallons Pi gallons yeah don't forget that Pi so when you multiply by pi times pi there's no decimal showing up don't think there are enough numbers to do it it's not like it's like was perfect number no it will be rational but 2 3 4 9 9 1 missed one two jeez 2 3 499 1 13 05 gallons that's 2 billion 2.3 billion 2.35 billion gallons of water it's lot of water want to check the accuracy of that I'm going to Google that what's it called the Great Blue Hole how many gallons of water are in let's see they say the ocean the world pool the Pacific Ocean the Great Blue Hole I've never heard of the Great Blue Hole they say approximately 2.35 billion gallons what was this answer yeah about 2.35 billion okay so they they they worked it out right obviously they probably got the factual numbers and then kind of worked everything out but isn't math great isn't math great okay number 36 like I'll get the answer I'm just going to look it up number 36 the volume addition postulate States the volume of solid is the sum of the volumes of all its non-overlapping parts agreed we just did that literally for like problems 31 and 32 or whichever use this postulate to find the volume of the block of concrete in example five by subtracting the volume of each hole from the volume of the large rectangular prism which method do you prefer explain your reasoning okay so mentioned this in the lecture portion let me let me pause and get this cylindrical block thing one second and we're back my gosh had mentioned just in previous problem actually had mentioned in think this problem that what the book probably would have done was subtracted this circle area minus this circle's area and then multiplied by height as opposed to find the volume of each of them and then subtract already have my preferred version it's what did find the volumes of them and then subtract just because have more tangible understanding of what it is that I'm using so we're going to do that one though forget what their answer was but we'll just we'll just deal so here we're going to subtract these two volumes now the volumes are the same so once find one of them you're all good to go but there's that one and hopefully can kind of copy and paste this and just say here's the other one so here's that one here's that one there so am allowed use this postu to find the volume of the block by subtracting the volume of each hole from the volume large okay they should have done the large rectangular but they didn't yet because they subtract the area of the bases so we'll just redo the problem so I'll call this object one and one of these object two because you're still going to be doubling whatever ever that is so the volume of the total for me whereas they would have done they would have done the area of 1 minus 2 areas of the 2 time height I'm going to do the volume of one minus two of the volumes of the other which obviously means you have to do the the base times height stuff still so I'm going to be using this one whereas the book used the other and prefer this one it's both are fine think the other is really intuitive think it's nice to just see that the area the base just feel like there are some things like what if we cut out pyramids right what if we cut out you know different kinds of things where Suddenly It's not congruent everywhere think that this one is more practical makes more sense anyway so the area of the base of the first one it's all rectangular so the area of Base 1 is going to be this 1.31 * 66 and they got number at the time don't remember what it is and don't know if they saved it till the very end but that's exactly something give me second calculator 1.31 time 0.864 6 square feet the the number the answer of this whole thing was really small it was less than one believe for the cubic feet even yeah cuz even the height is going to be remember the Height's the same for all of these by the way the height is 0. 66 ft so we can use that that's why they justified in doing what they did so the volume of object one is Point area base times height which is exactly and this is exact which is exactly 570 636 cubic feet okay there's object one the second object has an area of Base which is 39 * so 39 * 33 which is exactly this is all exact 1287 square feet this was five right so the volume of this is going to be 01287 time 66 which is exactly 942 cubic feet 0 8 4942 okay so the volume of the total we're going to take one of these minus two of those so volume total is 5706 36 xus 2 * 84942 and that's exactly calculator time. 636 - 2 * 84942 that's exactly 4752 cubic feet now they should have gotten the same answer from before let me go find it and show you let's find that page they got this one where it says 0 4 that's so they decided to round in the end but the 0 40 should be equivalent to what we just said before this cubic feet by the way so yeah that thing works out and should be equivalent to this one 4 yes so same answer as we got there all agrees we are in the right ballpark so I'm going to write the exact of Point 4752 just just justify that actually did it all where'd you get the other numbers from computed it so prefer this method as can organize my work more clearly for myself and it can work practically for all kinds of objects not just same types of prisms see this works because well sorry not just same type of polyhedrons like said if had so if had two prisms if cut out cu we cut out square prism if cut out triangular prism from there it would still work but if cut pyramid out from there or sphere SE semi hemisphere from there it wouldn't work and also if the heights were different you know don't want to be unique and specific to that kind so like being how lay it out here all right number 37 in writing both of the figures shown are made up of the same number of congruent rectangles explain how Cavalier's principle can be adapted to compare the are of these figures well this is kind of how explained it kind of said it was stacks of paper don't know what my official explanation would be in this kind of scenario other than to just state that you know they're the same because they have the same height the the base areas are the same so guess yeah I'll say that so they are the same they are the same as they have the the same height and well not just area base what did they call it where the the cross-sectional area they have the same height and cross-sectional area think of it as stacks of papers that's that's kind of where go with that one and yeah well okay take it back we won't call that cross-sectional area we would just say the rectangles okay because this this isn't so much this isn't so much 3D so I'll say the rectangles have the same length and the rectangles have the same length so maybe it's not saacks of papers yet because this is more two-dimensional in in Scope when we're looking at that one so that's what I'd say for that one okay there's number 37 then number 38 let me and 38 each stack of memo papers there you go contains 500 equally sized sheets compare the volume explain your reasoning this is where I'm sure could say the cross-sectional area that said before so I'm just going to copy and paste this guy and this time I'll mention cross-sectional area so yeah that's that's how see whoa that that's how see Cavalier's principle they're the same as they have the same height and cross-sectional areas that's that's where I'm going with that that just because you move it around doesn't change how much space it occupies where it occupies is different but yeah that's what like about saying that all right number 39 open-ended sketch two rectangular prisms that have volumes might need more room sketch two rectangular prisms that have volumes of 100 cubic in but different surface areas include dimensions in your sketches now even if you don't know how to calculate surface areas just basically make sure they're not congruent to each other let's do this break up 100 I'm going to break up 100 into kind of different factors here we could do like 20 * 5 and we could do 25 * 4 well no the five is still kind of there guess there aren't too many ways to break up 100 suddenly I'm realizing this 20 * 5 and 10 * 10 as long as the 10 that break up here can be like 5 * 2 and this 20 is some form of like 5 * 4 at least here these are the unique Dimensions that we can have of these prisms rectangular that's rectangular with this height so this can be like length width and height and this can be length width and height believe that these would then have different I'm trying to make sure these would have different surface areas even though you don't know how surface area stuff works necessarily let me let me think about this jeez 10 let's see 20 plus 140 is 160 and 20 no 40 plus 90 is 130 80 80 plus 990 is 170 okay at least these will be different don't know if that was the most unique way for me to have done it but that's my breakup of this thing so they said draw these two okay I'm going to draw one of them here and this has the height of five the length cubic inches 5 in and 4 in and 5 in and then this one here has height of 10 where it has length of five and width of two so this is going to look different you know like that so this is 10 in and 2 in and 5 in so they were close having the same surface area though when just kind of calculated them in my head but yeah so the volume here area of base is 5 * 4 which is 20 and the height is five so the volume is 20 * 5 which is 100 just like we wanted this one here has area of base of 5 * 2 which is 10 and height of 10 and the volume is 10 * 10 which is 100 and obviously it'll work out because the numbers that chose were from factors of 10 but that's just growing it so yeah they will have different surface areas that's that's open-ended so you can choose what you need to probably could done like square root of 10 things like that but that's okay all right number 40 making an argument your friend says the polyhedrin shown as triangular prism correct no triangular pyramid your cousin says it's triangular there it is Pyramid who is correct to explain your reasoning your cousin is correct your cousin is correct there are no parallel congruent bases opposite rectangular lateral faces or parallelogram faces like prism would have the triangle the Triangular lateral faces meet at an apex like pyramid friend is incorrect cousin is correct okay that was fast number 41 making an argument wait was that one making an argument as well yes it was so number 41 making another argument prism in cylinder prism and cylinder have the same height and different cross-section areas your friend claims that two solids have the same volume by Cavalier's principle is your friend correct to explain your reasoning no if they have different cross-sectional areas then then it doesn't work your friend is not your friend is your friend is messing up today your friend is incorrect if both prism and cylinder have volume as area base times height and the heights are the same but the cross-sectional areas as in the base areas are different then the volumes would be different now if the volumes now they won't be shaped the same but if the volume if the areas of the bases and cross-sectional areas were the same you would get the same volume out that's what Cavalier's principle was actually bringing up they have different base areas here so no if they had different heights it's possible all right number 42 Cavalier's principal this is the first time I've ever heard of the principal so it's I'm they're throwing lot of that at you Cavalier's principal States the two solid shbl have the same volume do they also have the same surface area explain your reasoning I'm trying to determine mathematically first first was trying to figure out logically I'm trying to ask myself that question now maybe what I'm thinking about is this whole length width thing length width height now the height is already applied and it's constant there but now when it comes the length width that length and width hasn't changed that height hasn't changed but this here is different number think this has more surface area I'm trying to think practically with paper think this this has more surface area because think these sides have more surface area because this now becomes hypotenuse you guys don't know too much about surface area yet so it's kind of interesting that they ask questions when it comes to surface area here think these ones are larger now think there's more surface to be covered there because that hypotenuse aspect so surface area is different the surface area of the oblique prism solid has greater surface area on the slanted sides because those Heights are now hypotenuses which are longer than the previous Heights displayed yeah whereas the other ones have the same area this this area and this area should be the same because they share the same base length and height the ones on the right side should be different this should be larger than that interesting okay all right two more questions number 43 barn is in the shape of pentagonal prism right there it's not regular pentagon though with the dimensions shown the volume of the barn is 9,072 cubic feet find the dimensions of each half of the roof okay it is pentagonal prism that means therefore the pentagons are the base and the 36 is the height so you have options you can either divide this into like rectangular prism and triangular prism like that and then add those two together or find the area of the pentagonal basee which think you'd still need to do that kind of thing you'd have to find the area of the two separate things and multiply by height so now I'm going back to my previous argument of saying would rather find the area of that base on its own and multiply by height or what rather find the area of the two different volumes the the excuse me the two different volumes and add them together this time think I'd rather do the base one I'm kind of more interested in working this out where this is one of the pentagonal es right here they're right angles they have this 18 this is8 this is some unknown which we're trying to find guess we're trying to find it and the volume's the nend total but this is split up into two different parts it's not regular pentagon I'm not doing 1/ half APM times perimeter take this area 8 * which is something 144 144 144 square feet and then find this one as well actually think I'd rather do the volume set I'd rather do the volume set because know the 9072 is separation of those two parts so the only problem is don't know it's it's isoceles there these are Cong but they show nothing else about angles they don't say it's like 45 4590 unless they say the top is right angle find the dimensions of each half of the roof interesting they don't say that's right angle they just show this as ioses they don't show any angles there which totally changes everything the barn print with Dimensions shown okay don't know which version I'll do yet let's let let's do the one where they're separate so this is object two and object one now there's volume as this thing sticks back right I'm giving it another another appearance so as this sticks back there's this 36 feet that's going to be the height that's your height because it's the distance between your two bases this base and that back base so the volume of object one is going to be area of Base times height which is 5,184 cubic feet now the volume of the total is the addition of these 2 V1 plus V2 whoops plus V2 and we know that V1 and we know the of the total the volume of the total is 972 so 9,7 2 equal 5,184 + sub 2 so sub 2 is 972 minus that 3,888 cubic feet now the volume of this is also area of base time height so 3888 is area of Base times hold on my mouse my mouse speed just changed times 36 so therefore the area of the base of volume 2 is going to be 108 Square ft what don't know is how in this triangle in this triangle with these two things as X's is even though if know this area 108 ft if know nothing else about it other than its isoceles triangles how can determine what is need an angle measure of some sort somewhere let me pause and think about if there's something missing if know this is right angle I'd be good if know this is right angle then know this is the of 108 it would be 6 < tk3 would be 6 < tk3 6 < tk3 hold on let's so 6 < tk3 would be 10.39 ft but don't know that that's right angle so don't know so let me pause and see if there's something I'm missing about angle stuff here give me second okay caught the issue and it's because went off my own draw should have gone off my own drawing more this is 18 so this is 18 so if that's 18 then right here so don't so that's no that's no known right angle I'm trying to erase it there we go so this is 18t down there okay that helps me lot more what do do still so if that's 18 the 108 let me move the 108 out of there guess it's married to it so basically need to find this height need to find that height and then use some Pagan theorem stuff right so the area of this basee is2 the base here 18 don't know how didn't see that the first time times your height the height of this so is 12et so have 12T height and if that's 18 then this is isos Le half of that will be 9 and this is 12 looks like Pythagorean triple of 3 4 5 so this will be 9 12 13 91 12 15 so x^2 = 92 + 122 is that setup we are going to get 15 here X2 = 81 + 144 x^2 = 225 = 15 ft so what do they ask for find the dimensions of each half of the roof it's 15t by was it 36 36 ft 15 here 36 there so that didn't right 15t by 36t there it is okay fun question and think this is the last one here number 44 wooden box is in the shape of rectangular of regular pentagonal prism the sides top and bottom of the Box are 1 cimeter thick the side so see that see they're talking about this part they're saying this part's 1 cm believe don't know about the top bottom that they're talking about approximate the volum because this part sticks up as well okay gotta got you gotcha approximate the volume of wood used to construct the Box round your answer to the nearest tenth so It's probably hard to tell I'm going to try and draw let me let me do this I'm going to pause it and get higher quality image of this one on my side of the thing okay as it turns out couldn't get it super high quality the image itself is little blurry still but let because wanted to separate this stuff and really show you this what they're referring to for 1 centimeter there's distance between these two height wise that's one cimeter distance and then the got to sneeze excuse me and then this distance for example probably got to pause again to drink water this for example is also 1 cm so 1 cm width there and 1 cm height off those things so when they're talking about the six here that is the top and bottom of the thing but then it's cut into it it's it's cut into it 1 cm and on the sides are cut into it 1 cm so we have couple things going on here and yeah let me let me pause and take care of this okay still little congested from the sneeze but better on the breathing so we have couple things going on here but the basic thing for me is what you see is the 4 cm and 6 cm the 4 cm is the apem of what would be wooden pentagonal prism without any hole stuff get into it and the height of the prism would have been 6 cenm so finding the volume of that will be part of what we do we're then going to cut out pentagonal prisms on top and bottom they said there's one underneath as well so we're going to cut out two of the same volume one right here and one on the bottom and in both of those instances they have an apem that's 1 cm less than four and their height is one 1 cm and so we're going to be just using those ones and subtract those two things out that's that's the idea that I'm seeing here I'm going to go ahead and go to work on that let me just shrink this little bit group it shrink it roll it pad it Mark it with okay so I'm going to deal with that object one object two stuff from before this is hard to hard to read this is object one and invisible object two for all intents and purposes invisible object two is the thing that's going to be cut out so the volume of the total it's going be volume of object one minus two volumes of object two give me one more thing got to take care of okay so for object one for object one we need the height the height of object one will be 6 cm that's the easy part the area of the base is going to be 12 APM time perimeter so we still got to find find the and the so let's do that based off our Pentagon we've done lot of regular pentagons here in the past but this has an APM of 4 cm we just got to find the perimeter by finding out this length here by dealing with the central angle we've seen it lot of times it's 36° half of 72 72 is 360 over 5 the of 36 is excuse me the tangent tangent of 36 is over our apem of 4 so is 4 * tangent of 36 so double to get side length you'd get 8 * 5 is perimeter that's 40 so 40 * tangent of 36° is that guy so your volume of Base 1 is 12 apem time perimeter now I'll leave it exact as long as can simplify that half of 40 is 20 * 4 is 80 so you get 80 * the tangent of 36° so that is the area of Base one so as far as full volume of object one we're going to be doing area of Base times height and the height is six so 6 doesn't multiply 36 it multiplies with 80 so volume of object 1 is 480 * 10 tent of 36B CM believe that that's all kind of panned out properly for that that's object one now object two is smaller pentagonal prism with not too much got to change here really honestly we just got to make sure that we can follow the same patterns we had before the difference is this is going to be 1 cm less of an apam which means it's going to be 3 cm same 36° and all that kind of stuff when it comes to this that's right here cuz this is still 72 36° the here let's call this X1 and X2 Xmen United X2 is well 3 * tangent of 36° 3 * tangent of 36° honestly think here could almost do the like ratio bits so the perimeter of object two is going to be double that and then Time 5 30 30 * tangent of 36° hope that I'm getting my work all sped up properly with that that should be the perim of this so the area of base 2 is 12 APM time perimeter which simplifies to 15 * 45 time tangent of 36 degrees and the height of object two is 1 cm so this will also be the volume the volume of it because times one area of base time height is going to then be volume with that when put together that's cubic cm so the volume of the total whoops volume of the total we're going to be taking sorry we're going to have some spacing out here this is kind of the last part we're going to be taking the 480 tangent of 36 480 time tangent of 36 minus 2 45 * tangent of 36 is because there's one on the bottom of there as well so that's 480 minus 90 notice they both have tangent of 36 so it's like like terms so it's 390 * tangent of 36° in cubic cm with the calculator we will get 390 * tangent of 36 around 28335 I'm going to verify that everything kind of works out the way want it to don't always trust my work it's one of those things I'll probably pause and check it out because it's one of those things where one slight thing would have gotten wrong and go shoot have to change everything but let me kind of take look at everything and explore it and make sure that all is how the problem was supposed to be suggested and all that but there's my answer for now hold on okay looking at the problem more so think calculated everything wanted correctly but looking at the problem more was asking myself why the box would be constructed this way if it's box and it looks like something you can open up and put things in the Box would be filled with wood so was asking why the box would be constructed that way and the answer is it wouldn't the answer is it would be hollow inside so started thinking again that these didn't look the same this doesn't look like 1 centimeter like that does as well don't think this part part don't think that this part's 1 cm don't think this counts as certain height thing don't think this is what they meant when they said it's 1 cm thick just like this part here think this whole top is just whole top what 1 cm think is this right here that it's cut into 1 cm on top and bottom so it's thicker like that so you're not cutting out from the top top and bottom you're cutting out from the inside and there's still 1 cm on top and bottom so the one change need to make is the height of volume 2 is not 1 cm it's 4 cm if there's 1 cm removed from top and bottom then one cm removed from six is four so this changes from well now should write what this thing is area of base two is now this 45 right but we have to now multiply that by four for volume of the second one so volume of object two will be area of Base times height so think this is supposed to be 180 * tangent of 36 and that's what we remove we remove the inner volume of that inside guy and it's hard to kind of draw unless have like chest thing to show you so don't remove two of something just remove that something so remove 40 180 excuse me times tangent of 36 think that's different number now cuz now that's for so now it's 300 300 * tangent of 36 and then we'll get different value there because was sitting there thinking like the numbers are good but and then started looking at the thing and going wait what is this object right want to put stuff inside of it otherwise it's literally just box with nothing to do and then was asking myself the whole time that doesn't look like 1 centimeter with the other thing and think they think it's photo so don't think that would make sense beyond that so when calculate this one 300 * tangent of 36 I'm getting 27.96 so what do they want to surround to nearest 10th so 27 28.0 28.0 cubic cmet so that's where I'm going to go with that one that's what believed it's supposed to be based on the box nature the appearance of the thing and the nature of boxes themselves we're remov moving from there the sides top and bottom are 1 cm thick that means the inside is hollow that's what we're removing the height of four not one and not two of those ones all right guys that to do it for this one this is Mr Robinson thank you so much for watching and next will come 115 116 117 there's still volumes of things think pyramids come next and then we hit surface areas of them and then we hit volumes and surface areas of spheres lot more to go with lot of these calculations one little thing can screw lot of things up but hopefully you followed along and it all make sense take care I'll see you in the next one