Volume of Pyramids and Cones practice problems

well hell welcome youtube mr robinson back here with you another brand new exciting video on math based of course and as always it is an honor and privilege to be serving you as it is every day here in my virtual classroom step on inside we are going to continue our talk about volumes of different solids the other day we talked about volumes of prisms and cylinders today will be about pyramids and cones now when do my practice problems out of the textbook pages that do these are actually two separate sections so when do practice problems i'm actually going to jump around and try and make sure that do some problems out of the pyramids and then some problems out of the cones they don't really intertwine unless maybe with composite figures when we hit cones maybe there are things with pyramids don't think there were i'm not too certain but we'll see if they come to be now if this is your first time kind of venturing in this area want you to know couple things number one believe there are about 18 practice problems that i'm doing so this can really go one of two ways which i'm airing to the side of probably this being kind of longer video but given that most of these are kind of based off what we learned from volume of prisms and cylinders and you'll see why in the beginning i'll teach in early part and then i'll do the problems but based on that notion want to do little more drilling kill as in like really kind of just do the problems for you every so often they'll ask question of different kind of variety where it lets me teach something else little different but if it really is just finding the volumes of them if you saw what did in the previous prisms and cylinders video then that layout will be very very similar so really do urge and ask that you take look at some of the layout in that video if you do get have confusion here if you do get confused on some of the things that we do all right now like said want to teach concept kind of laying into it so you understand the principle of what's happening and then we're going to be doing them but 18 practice problems combined of pyramids first cones after and i'm really circled problems that want to do so basically what's going to happen is we're going to look for and find those problems that we want so let me get started with it right here let me get there and you know it starts in this like number four problem here i'll do these four and five with the diagram most all of them if not all of them have diagrams but want to start with this thing right here this is based off previous problem but want to start with this and talk about how we find the volume of pyramid and then i'll think there's probably an image very similar to this for cone if you notice there is pyramid kind of trapped inside this prism now if you imagine both of these share the same don't know if it's square base or whatever but both these share the same base like they take up the same amount of space on the bottom and both of them ideally have the same height as in this length right here and this length right here from that vertex all the way down here these are congruent to each other that and that would be the same length so imagine pyramid that fits exactly inside prism that has the exact same base area in the exact same height did you know that the volume if your if the volume of your prism was area of base times height that the volume of pyramid with again the same exact properties that we're referring to with regard to area base and height it's exactly one third of that that's right whatever the ideal prism would be with certain area of base and height that you would do the pyramid that is equivalent to it in that exact same sort of way would take third of that space you would divide it by three so when we do number four i'm not going to do this for every problem but when we do number four if drew prism that was equivalent equivalent right triangular prism that you know looked like this right here so i'm trying to make sure to get right right if there's prism triangular prism that looks like that whatever that volume is this is going to be this pyramid is third of it and that's what we're going to do we're going to take the area the base we're going to take the height we're going to do exactly what we did before and then divide it by three now i'll tell you what they are gonna ask you to round your answers to the nearest tenth every so often might well was gonna say was gonna leave answers as fractions but they're asking us to round them to the nearest tenth haven't done these problems yet but will imagine that want to use my calculator especially with decimals such as these ones right here i'm not going to dare play with them and get too dangerous with them just because there are 18 problems to do so let's get started number four find the volume of the pyramid have right triangle as base here as as in base of the pyramid have right triangle as base right down here let's find the area of that guy by if establish this 12.5 millimeters as the base of the triangle in and of itself the height is perpendicular to that base at 15.2 so if we do the area of the base of the pyramid of the triangular pyramid by the way you name pyramids based on their bases like you do prisms this is triangular pyramid the area the base is one half the triangle's base length times its height so have to do half of 12.5 times 15.2 now think for the sake of time guys you're going to take my word for some of my calculations here but i'm going to do 12.5 times 15.2 on my calculator and then i'm going to go ahead and divide that one by two it's clean number what do you know so that actually gives you 95. so that number is take my word for it that's 95 square millimeters every so often i'll go to that calculator for you to see final answer especially if have to round but just for some of these just want to kind of do it otherwise have to flip buttons and you know it's kind of big back and forth the height of this pyramid is 8.1 8.1 millimeters so the volume of the pyramid you're going to be doing area base times height that would be the volume of prism with that thing but now we're going to divide it by three so one third of area of base times height and we'll get something by the way before continue just want to let you know did put it in the description of the previous volume video but there is mistake on my number 10 in the previous video where accidentally if you didn't watch it you have no idea what i'm talking about but were you where added two volumes and composite figures together instead of subtracting them that was my bad if you take look you'll kind of know what i'm talking about should have subtracted them i'm just kind of calling it out because remember that it's something all right this is exactly 256.5 cubic millimeters right there so that is going to be the volume of the pyramid yes already took the third divided by three if you want to take look at this one here got those ones right there so that was me doing my 12.5 plus 15 point or times 15.2 divided by two that's taking half of it and then took third of 95 times 8.1 and got 256.5 cubic millimeters all right so there is number four let's keep moving forward with number five same kind of instance you know pyramids this is was going to say square it's rectangular pyramid here so area of base we're going to be doing first here area of base is going to be 6 times 4 which is 24 square inches the height and remember guys height is still perpendicular it's the it's the distance up to that that vertex forget what they call it apex vertex crown it's the height up to that point where all these things meet here and it's the perpendicular distance that we find so 17 is that established height right there that is the height of our pyramid like prism again if you drew it would be that height from base to base so this is 17 inches right there so this one's actually guess easier than this one the volume as far as the numbers are just right there volume is third of 24 times 17. this one want to do by hand because i'm gonna take third of 24 that's 8 times 17 actually don't know that one 8 times 20 is 160 minus 24 is 136. so this is going to be exactly 136 cubic inches on that one okay so there's something you can do by hand guess and there's that one all right numbers four and five let's just keep moving forward any problem that circled is one that want to do this one does not have drawing with it it says find the volume of hexagonal pyramid with base area of 25 square feet and height of nine feet so once again straight up guys well here i'm going to draw kind of an equivalent one hexagonal pyramid guess would look like that right there and then all points meet to the top we okay so there's your hexagonal pyramid the base area is 25 square feet the height is nine feet so the volume is third of 25 times nine once again want to do this without calculator here can't take third of 25 and get an integer but can take third of nine and that's three three times 25 is 75 so here we'll be getting 75 cubic excuse me cubic feet on that one arguably easier than the others just no drawing unless you want to make one all right have another is this it's another hexagonal pyramid this time with drawing the area of the base of the hexagonal pyramid is 24 divided by tangent 30 square centimeters we saw something like this kind of with the prisms with the kind of weird look find its volume this height is 4 root 3 centimeters this base length of this base edge length of 4 it's kind of just throwing it out there just to kind of throw it out there for you it's not really doing anything i'm not going to use it because already have the base area so i'm good on that front got my 24 over tangent of 30 that need for that part i'm going to multiply that by this 4 root 3 right here and then i'm going to take all of this and i'm going to take third of it i'm going to divide it by 3. so the volume is 1 3 the area of the base times the height like that and this is where do want to go to the calculator can take third of 24 yeah but got to do the tangent of 30. got well actually no you know what you know what know this answer the hold on i'm gonna i'm gonna try this one in my head and then i'll see if it works so 8 think the answer is going to be 96 okay so let's see because know what tangent of 30 is exactly so here we go let's go to the calculator and check it out where's my graphic calculator so we're going to do one third of you know what need to make sure i'm not want to make sure i'm in degrees here every time use this calculator it resets to radians so 1 3 of 24 divided by the tangent of 30 got to close this parenthesis and then this parenthesis and then times 4 root 3. i'm expecting 96 as an answer all right so we're getting don't have to close that but will so 96 there it is so we're getting nine you can't see it my face is in the way let me press enter again 96 96 96 96 so there it is 96. okay let's go back and that's going to be volume of 96 cubic centimeters there all right there we go all right sorry got texture okay okay now we have some composite figures we i'm going to do 8 9 and 10 here let me see if that's the and then got number 12. so have four more pyramid questions okay so eight and nine or two composite figures here we have rectangular no well it's cube we have cube and then on top of it we have square pyramid say that because this was square base right here and that's square base now the pyramid is not squarish in its height as in it's not 12 centimeters up but still square pyramid by the name of the base so let's call this one figure one and this one figure two so can find the volumes of them separately now because this is cube right here if this was 12 this is also 12 up here so this pyramid occupies the same base area you know both of these occupied occupy the same base area this base and this base are both the same thing so you know what i'll just put nice big old capital for both of these things capital here it's 12 by 12 area of base for either of these and that is 144 square centimeters for either of them for the pyramid or the cube okay now the volume of the pyramid excuse me the height of the pyramid is 18 centimeters so the volume of the pyramid here is going to be area one-third one-third area of base times height there's probably one of those ones should do with the thing 48 times don't know 48 times 18 6 times one four i'm using calculator there so 18 times 144 divided by three don't want to linger too long for you guys 864. so we get 864 cubic centimeters here now again did composite figures for the prisms and cylinders talk so if this is new to you basically what i'm going to do guys in composite figure get to add both of these together because it's just one volume on top of the other so the height of the second guy here was 12 centimeters so the volume of the cube here it is literally it's 12 cubes but as far as area of base times height goes 144 times 12 used to know what 12 cubed was let's see cubed 144 times 12 we're getting 17 28 maybe didn't that doesn't sound familiar 1728 cubic centimeters so the volume of the total thing here total when you add up both of those v1 plus v2 right and this time we do add them this time because it's one on top of the other that's going to be 864 cubic centimeters plus 1728 cubic centimeters which is don't get this one wrong 25 92 is that right 17 28 plus 864 25.92 so 2592 cubic centimeters there and that is the volume of that whole composite thing right okay this one looks little you know it looks tough as far as like man there's lot of stuff there so let's break down what we got we have two pyramids both on top of this rectangular prism okay each of the pyramids are 7.5 centimeters in height now there is don't know what guess the rectangular pyramids as well we have 12.5 centimeters not just right here but also right here so we do know that length is 12.5 now here's the thing we know this whole thing is 25 but we don't know what each of these lengths are right here you know what it says they're congruent was going to mention something about that was gonna say that we don't know what each of these are but you know it honestly doesn't matter if this one if this was 12 and half and 12 and half or if this was 15 and 10 or if this was 1 in 24 the principle still lies the same way with the volume of these pyramids but the fact that these pyramids are congruent really helps it it does it helps us talk about this lot faster so here's what i'm going to do guys i'm going to call this one figure one i'm going to call this figure 2. i'm not going to give this figure name because the volume of one of these is the same as the volume of the other if you want me to call this figure one as well can but the point is one of these i'm just going to double if that if that sounds good so in figure one we are let's let's start with this 25 over 2 is 12 and half what i'm pointing at there is this length right there this is 12 and half because this whole length is 25 these are congruent by those marks we are getting 12 and half from that all right so 12 and half from that we are going to looks like it's square pyramid then so we're going to go ahead and find the area of the base of each square pyramid so the area of base 1 it is 12 and half times 12 and half so 12 and half squared do not know that number that is 156.25 156. they say round to the nearest tenth because that is the final answer something else that mentioned greatly in my last video with the other things the height of this one is 7.5 now remember when find the volume of this ultimately will need to double it for the final answer so the volume the total volume of each of these pyramids here is 156.25 times 7.5 which definitely don't know what that is so that times seven point five we're getting one one seven one point eight seven five once again we are not yeah it's on five we are still not yet rounding this may be volume but it's not the final volume we still have to double this and we still have to add it to the rectangular prism down below all right number let me change these colors here so the blue doesn't bleed with the other question that we're looking at there let's get some purple here all right the second one the rectangular prism here the area of the base well that's different purple the area of the base is 25 times 12.5 now that should be double the area of one of these bases here because it's that plus that so it should be 312.5 but just to make sure that didn't do any miscalculations 25 times 1.2 or 12.5 that's 312.5 so 312.5 square centimeters and then the height of this one is 5. you can see that right there the height of this is 5 centimeters so the volume of the whole thing we are going to do 312.5 times 5 there don't want to screw that one up but i'm thinking 15 62.5 3 12.5 times 5 is 1562.5 so we got 15 you can't see that i'm sorry 15 62.5 cubic centimeters now we're still not done so don't have much room i'm just going to kind of fit it in this little region right here i'm going to box this out and we need to find the volume of the total so the volume of the total need to take need to double one one seven you know didn't take third of this was wondering what was going on there one third one third see got it got to be careful got to be careful i'm very sorry let me kind of keep going you might have been scratching your head saying what's happening there was wondering why that number was so big so that's going to be yo my bad point eight 1171.87 five point eight seven five divided by three so that's 390.625 so hey guys let me erase some of that really quickly there it's my bad okay something knew something was up just didn't wanted to wait to get the other answer before actually questioning it and now got it so 390.625 good thing didn't round early either needed to keep that number all right so we are now we're now going to be doubling instead 390.625 hold on one second just got message for some okay i'm back sorry okay so double 312 0.5 and then we're going to add one five six two point five so here we go sorry not three twelve point five three ninety point six two five three ninety it's this one okay so 390.625 times two plus one five six two point five all right that'll be two three four three point seven five rounded to the nearest tenth is two three four three point eight so two three four three point eight cubic centimeters there we go cubic centimeters there's the big guy yeah prone to make mistakes here every so often really got to treat that one third remember got pyramid there okay number 10 given square pyramid with height of 21 feet and volume of 3969 cubic feet find the length of one side of the square base rounding the nearest tenth so this is different kind of question especially without the drawing here we have square pyramid so do have to square down below right here and then they all meet up say like at top point like this so we're looking at you know this kind of thing guess that one should have dotted lines right there there we go all right so square pyramid like that kind of kind of off looking but the total volume we already know now this is three nine six nine now remember the volume is excuse me it is one third area of base times height so one third area of base times height is three nine six nine don't know if there's reason for me to draw the square we'll see there is because we need to find the side of square base they know that the height is 21 so one third of area base times 21 is that so there's some algebra to do here right i'm going to i'm kind of trying to get by itself to start with third of 21 is seven so got seven times is three nine six nine and then divide both sides by seven you get don't know what the number is three nine six nine over seven let's hope it's clean one furthermore let's hope it's even perfect square 567 don't know if that's that doesn't sound like perfect square to me so 567 so that's the area of the base they're asking round to the nearest tenth see they're saying find the length of one side of the square base well the base is the area of the whole thing because it's square each side length is the same so if treat these as x's this is what the drawing is for if multiply that by itself get squared so squared the area of the base is 567 and that's where you get to take the square root of both sides to find the length of just one side and the square root of 567 for you guys will show you this one on the calculator here let's go to calculator so it's it's this number here so got three 969 divided by seven to get 567 square root of that 23.8117 so around in the nearest tenth is 23.8 about 23.8 this is cubic feet 23.8 feet approximately so go back let me give the approximate symbol there as rounding measure there's an exact answer here's rounded one all right there you go now let's do number 12 same-ish kind of question here you got another square pyramid this time we got to find the unknown height the value of is height here so one same thing guys you have 200 cubic centimeters for the volume now volume of pyramid is 1 3 area of base times height the area of this base is 10 times 10 which is 100 square centimeters so i'm going to go ahead and substitute 100 into here we take third of hundred which i'm not going to do on the fly i'm actually going to do this little differently i'm going to is our right let's turn that now substitute that and call that let's let's multiply both sides by 3 to get rid of that so that's 600 and then let's divide both sides by 100 so 100x equals 600 divide both sides by 100 you get equals six so six is the height that is six centimeters and i'm set to jet that's it i'm good to go all right that is the pyramid-based questions that wanted to cover here guys so hope that that made sense and the stuff that we talk about with cones is going to be very akin to cylinders the same way that pyramids were to prisms i'll find figure to kind of talk about what it was that we did before some more drill and kill some more composite figures and think there might be word problem at the can't remember if circled the word problem or not maybe didn't either way let's go find them all right let's see moving on forward guys we are going to be looking at cones let me quickly scan this to see if they have the equivalent of what wanted to find okay this will be good enough even though we're going to do this problem later this is an inverted cone you know i'm going to draw my own because don't want an inverted cone so let's say had cylinder guys let's say had cylinder that looked like disa and then let's say had cone right inside of it you know cone cone is it's like circular pyramid it's like circular pyramid it's out of cone right inside of it same thing as the pyramid to prism pyramid is exactly third of prism's volume that occupies the same area of base and height same thing here area of base that circular base and the height that reaches up to here like it does here the cone is exactly one-third of the cylinder's volume so once again if volume of cylinder is area of base times height pi squared in fact pi squared times height because this time it's always circle then the volume of the cone is one third of that area of base times height which is one third pi squared times the height now i'm not going to be doing the formula like that but if you ever see formula like that you type that into google they'll say you're looking for volume of cone so they're going to be doing that kind of thing with you but i'll be doing area of base just like did from the previous stuff and identifying radii wherever can all right here we go guys where we are 26 minutes in let's see if we can get this done in under 50 minutes i'd love to but there might be more cone questions find the volume of the cone round answer to the nearest tenth numbers two three and four are all looking like that they all have different kind of properties though got an inverted cone with the base on top so the area of this base right here it is pi times the radius which is 1.9 squared got square 1.9 don't even know 19 squared three don't know 361. so it might be 3.61 1.9 squared yeah 3.61 so i'm getting 3.61 pi square millimeters for the area of the base of this cone the height is 4.2 millimeters so the volume of the cone is one-third the area of the base times the height which won't dare try and do in my head not that it's impossible but just don't want to watch it be clean number though times 4.2 and then take third of that and i'm getting 5.054 let's take look at that i'm getting 5 press enter ignore that one just did 5.054 5.054 so rounded to the nearest tenth should be 5.1 okay 5.1 does that sound right guess so because it's third of that thing so 5.1 hope did my math and all that stuff that was talking about correctly pi no something does sound wrong i'm sorry about that 3.61 pi there we go something seemed wrong about that value it seemed little small so let's go back there that is let's just let's go back let's do 3.61 pi my bad this is me working all day on math stuff y'all times 4.2 all divided by 3. there we go that looks lot better so 15.95 like that one lot more good thing was second guessing myself there 15.9 square cubic millimeters there all right all right this cone right here now what we're looking at with this 5.9 is indeed the diameter if we want to find the radius right here we got to take the diameter 5.9 and divide it by 2 and that's 2.95 2.95 feet so we have that radius right there we want to make sure that we get that before we square it we can't divide by 2 afterward that's already too late so the area of the base is pi times the radius squared 2.95 again leave this exact right here this is 8.7025 and there's the pi don't forget the pi square feet we shouldn't forget pi we all love pi okay the height height is not yet established now take look at this one guys this is the slant height of this thing at 6.3 we need to get the height we need to figure out what this thing is because height is always perpendicular here now this one gets little interesting there's kind of lot of work to be had here on number three because we have to do pythagorean theorem not only on decimals which is just cruel thank you book but also with this number that's even kind of weirder looking decimal so remember this this is our right triangle right here this is our right triangle right here we have the hypotenuse and we have 2.95 as this leg length right here we need to find so our pythagorean theorem would say squared plus squared equals squared okay we do know what 2.95 squared is though it's 8.7025 so that's good so we got 8.7025 plus squared equals don't know what 6.3 squared is it's going to be little under it's going to be around four that's four forty meant 39.69 all right subtract yeah definitely hopefully you guys have calculator and hopefully you're okay with me not showing the calculator on these minus 8.7 so we're getting 30.9875 and i'll let you know something right now let me first take the square root and see what get but then i'll tell you what i'm going to do yeah this number's bad okay let's look at the calculator here for one second guys so have 30.9875 this is just great problem isn't it and then take the square root and then get this number now we've talked about not rounding early we're not just finding the height we're finding the volume right so we have to use this height as an exact form in order to get there if you were handwriting this what would say is just leave square root of that symbol there if you know how to use this symbol for future prop for future part of problem you can store the previous answer and use it but it has to be exact so for now for you guys i'm going to leave the square root of that previous symbol thing here i'm just going to say for now that is the square root and hope i'm doing the problem right in the sense that hope didn't leave something out when was looking at all this that this is as obscure as this problem is getting but is the square root of 30.9875 i'm going to leave it like that right now just that's an exact answer that do have these decimals are exact to it all four of them and we have the space here so the volume is going to be one-third area of base times height so volume is one-third area of base that's 8.7025 pi times height square root of 30.9875 so i'm going to compute all of that right now give me moment to do that so that height times that base divided by 3. third is dividing by 3 there now i'm getting 50.73 something something so 50.7 volume is about 50.7 hope you're okay with me again not showing the calculator 50.7 cubic feet there i'm hoping that's correct can take look at the answer key stuff later but you know kind of follow along with what was going for we did have to establish that height this is of course leading to longer video in general but we did have to establish it because slant height is not the same as height slant height will matter when we hit surface area and of course slant height was important for using pythagorean theorem on this problem okay all right here we go so number four hopefully look this number is lot cleaner with stuff so on number four here we don't have the radius at all so finding area of base isn't that easy to find there we do have the height though it is 20. right is the distance from the base to the and the top it's tipped over now it's party hat that someone's like fun party tipped over this 22 right here is that slant height right that's the hypotenuse if you will of right triangle where you have leg right here and the other leg would be the radius right here so if want to find the radius length i'm going to use pythagorean theorem on this stuff here so kind of have have so much room here squared plus squared equals squared like that 20 squared is 400 22 squared think is 484 i'm going to go with my gut instinct on that one 484 subtract 400 from both sides and take the square root of both sides now that's not perfect number so just like the previous problem i'm going to leave that square root thing there if you really want to get fancy with me you could simplify the radical but we are going to be looking at so that'll be 2 root 21 i'm gonna leave that we are however looking at rounding to the nearest tenth anyway so i'm going to leave the square root of 84. like said in the previous problem here i'm not going to approximate this thing early on need to keep it actually this is this is perfect this is actually perfect don't even need this square root of 84. we need to find the area of the base right so area of base is pi squared we already have squared it's 84. squared is 84. the square the square root of 84 squared is 84. so pi times 84 there it is that's 84 pi perfect that worked that was cool that's square centimeters so the volume of the entire cone here it's one-third area of base times height you couldn't you can take third of 84 that's that's 28 right but we need the pi anyway so pi sorry i'm not showing the calculator here times 20 divided by three that's going to be 1759.29 so 1759.3 cubic centimeters there is your volume that's how much you can fill that thing up with all right i'm always questioning you know did did get all my numbers in properly don't know i'm gonna hope so would love to have teacher addition in front of me and see what the answers are but i'm kind of trusting my own math on these and if you see anything different and question anything different of course always leave comment right leave comment all right five and six these are oblique cones well yeah they're both obliques these are oblique cones they're not right cones as in they don't go straight up symmetrically to the top that's like right in the center here that doesn't change what we do with the height thing the height is still established here nothing's changed your diameter here is 24 inches so your radius is 12 inches so the area of your base here is pi squared which is 144 pi square inches okay that's the area of the base your height is 30 inches so your volume is one third that area of base pi can't leave that out again times height 1 3 of 30 is 10. let me give you an exact answer first 10 times 144 is 1440. so this is 1440 pi cubic inches and then i'm going to do 1440pi there in the calculator that would give us four five two three point eight nine so four five leave your answer in terms of pi look at that i'd even see that so leave your answer in terms of pi there it is so that's an example of leaving in terms of pi right we leave the pi there we don't use the 3.14 multiplication on it we're good that was the only one we're supposed to do that with so far right the other ones were round to the nearest tenth hope so all right this is further oblique but this is established as height because it is the vertical perpendicular distance from the base even if it's not actually at the base but from the base to the top that's how high one is over the other okay so 41 is the height that's 41 meters the area of the base is pi your radius is 9 squared that's 81 pi and once again we are going to leave our final answers in terms of pi let's kind of not forget that so that's square meters the volume is one third area of base times height so third of 81 is 27 and don't know 27 times 41 can get super close to that one at one you know think i'm gonna get 1 107 pi is that right 1 107 pi let's see one third of 81 times 41. i'm not multiplying the pi with 1 107. i'm not multiplying the pi with my calculation when do that because need to leave pi right here that makes sense need to make sure that kind of have it in terms of pi i'm not going to multiply by pi within my calculation so 1107 pi cubic meters okay we have composite figures for seven and eight let's see how many questions have left have my gosh composite have five questions left here we are at 38 minutes let's see what we can do looks like these are hollow points don't know specifically what's what sometimes here but we round your answer to the nearest tenth we're kind of back to that again so it looks like what's happening is this is the hollowed out part of this cone it's kind of like really thick cone like an ice cream cone kind of you know like like like shaved ice kind of cone it's like really thick though right super thick there's probably better it's like it's like thick funnel that has no out out point all right so figure one would ideally be the entirety of this cone even if this part wasn't cut out just the entirety of the cone and figure 2 is going to be the part that we cut out we are going to subtract this time and have to remember to subtract by writing out that the volume of the total is going to be the volume of the big guy minus the volume of the small guy so we're going to find both of these separately and then we're going to subtract in the end round to the nearest tenth now the heights that are established here we have 12 inches for the height of the big cone the height of the small cone is six inches as you see right there right there four inches for the radius there eight inches for the radius right there so area of this base is going to be pi times 8 squared which is 64 pi square inches and then this one is going to be pi times 4 squared which is 16 pi square inches you might notice that even though you got half the radius you got quarter of the area so yeah area area ratios are not the same thing as length ratios and volume ratios are not same as area ratios kind of something to talk about later i'd say all right the the volume of one is going to be one third area of base times height and remember i'm going to keep an exact answer to begin with because got to do the subtraction afterward and want to make sure not to round early for those things so that'll be 1 3 of 12 is 4 4 times 64 is 256. so this is 256 pi cubic inches for volume one volume two this is getting me kind of color blind is it for you guys well you're going to use pencil you're gonna have to deal with it let me just divide that volume 2 is 1 3 area of base times height third of 6 is 2 2 times 16 is 32 so that's going to be pi cubic inches now when you find the volume of total again we will subtract so volume total here that is going to be 256 pi minus 32 pi now just like variables if this was an we could combine like terms but it's pi you know we can combine like terms here because pi is common multiple of both of these so in subtracting these two i'm going to get 224 pi cubic inches that's in terms of pi the rounded answer after multiply that thing out 224 times pi is going to give me 703.716 blah blah blah so 703.7 cubic inches that's to the nearest tenth there are the composite figures there all right definitely some work to be had on that all right number eight we have an inverted cone inside of cylinder both the cone and cylinder are exactly the same thing now some interesting things can say about this one so we already talked about how the cone takes up exactly the third of the volume of that of cylinder so this is third of it which means every what think is happening guys is think that this is the cylindrical part and think we're cutting the cone out what what that means is everything we're trying to find here is two-thirds it's two-thirds of the cylinder so if you want to do this fast you want to try that you want to do this in fast way we could just well you know what i'm going to do i'm going to confirm that that's it but you can take two-thirds of the cylinders volume and that'll be your answer you'll be done you don't have to take the cylinders volume and the cones volume and stuff good news is they have the same area of the base and they have the same height think that's another way of showing that it's going to be two-thirds so the area of the base for both of these is going to be pi squared so that's 36 pi square feet the height for both of these is 10 feet so the volume of this cylinder i'll just call it still this time the volume of the cylinder is going to be area of base times height which is 360 pi cubic feet and then the volume of the cone is one third of that so let's just say one third of 360 pi one third of area base times height that's 120 pi cubic feet guess this problem is not too long anyway the volume of the total we're going to subtract the volume of the cone from the volume of the cylinder so we're going to do 360 pi minus 120 pi which is 240 pi and 240 is indeed third of we you know we still to round this but 240 is third of 360. two-thirds of 360. and yeah so that's you know that seems to make sense that that's the answer so the volume approximates to 240 pi we're looking at you know 753.98 now if you round up check this out 753 753.98 cubic feet but if want to round to the nearest tenth need to kick the nine up to zero that means have to turn this into four that's going to be 750 excuse me 754 54.0 zero cubic feet right there that's rounded to the nearest tenth and yeah want to put that point zero want to make sure that i'm establishing that precision that that was actually portion of it when kicked that thing up okay three more problems here we go cylinder with cone on top of it we got 13 look at this 13 meters is the entire height this entire thing right here so you chop off two meters there you're left with 11 meters as the height of the cylinder okay i'll call this figure one figure two 11 meters the height of the cylinder and then so that 13 can almost be ignored at this point you know and we have think this is one meter radius which means this also has one meter radius right there so the area the bases will be the same for both of these guys the area of the base is pi times one squared one squared is one times pi is pi so the area of the base of these unit circles if you will is pi meters that looks really weird if you want me to say one pi later could but it's pi meters it's pi meters the height of the cone is two meters so its volume is going to be one-third area of base times height these numbers are really weird so two-thirds pi cubic meters okay the volume of the cylinder is area of base times height which is 11 pi cubic meters are we around guess we're still writing to the nearest tenth it's the same instructions so the volume of the total is two-thirds pi plus 11 pi and it's it's always with these things i'm always thinking i'm doing something wrong now because there's only one little mistake that can these are the hard ones for me to grade for students because they are always going to get something wrong if i'm going to get something wrong sometimes so are they especially so it's always hard for me to award partial credit and stuff because they mess up somewhere and you know we got to kind of look for it all right so two-thirds pi i'll just do two-thirds plus 11 of course that's 11 and two-thirds and then multiply that by pi here by pi so we're getting 36.65 so 36.7 36.7 cubic meters that's the volume of that composite figure how much time we have 47 i'm not going to get in 50 minutes didn't know what to anticipate for this and probably the most confusing drawing of them all it's basically cone with cylinder cut out of it how tall is that cylinder that's some work we gotta play with here all right that's that's something to figure out let's let's see how we can crack this one okay there there's more than one way to do it but let's let's seek those ways all right the one assumption we're gonna have to play there for this to work is that this has symmetry to it meaning this is right cone and right cylinder otherwise this doesn't really work and it kind of better be with the 3 going there so if this is 5 then this also must be 5 right here since the whole thing is 10 that means this part is 5. and i'm talking about this right now because need to find the height of the cylinder unless have it don't don't see it so unless have it so that's 5 right there okay now if that's 5 what i'm going to do is i'm basically i'm constructing this right triangle here i'm going to find out what this length is right here this this little i'm going to find or yeah in front of this little right here by trying to identify all this furthermore we actually we had to find the height of the cone didn't even notice that we got to find out that as well so we use pythagorean theorem couple times and there are other ways you could find out what that is like for instance imagine this is half of this for similarity you know purposes but will still do pythagorean theorem kind of concept on both of them to figure it out all right the whole diameter is 12 so half of that guy is 6. we know that this length right here is 10 as the other one was 10. so have some h's i'll call this h1 i'll call this h2 should just call them green and blue okay h1 or the pythagorean theorem and know the answer already should be 8 here but h1 squared plus 6 squared equals 10 squared so that's h1 squared plus put in parentheses otherwise that looks weird plus 36 equals 100 h1 squared equals 64. if you subtract 36 from both sides and take the square root of both sides you get sub 1 equals 8. so there's that 8 that was talking about there so now we know that the height of the cone is 8. now imagine this one's going to be 4 right here how do we get the pythagorean theorem part because we need to figure this out the whole thing is 12. they established the radius bit here is 3. so if this is 3 and this is 3 then that whole thing is 6 in this circle so all that remains is another 6 outside of here because the whole thing is 12 so divide into 3 and 3. so that's going to be 3 right there little confusing right so we have as leg and 5 right here so 3 squared plus sub 2 squared equals 5 squared 9 plus sub 2 squared equals 25 subtract 9 from both sides you get sub 2 squared equals 16. take the square root of both sides you do get sub two equals four these are by the way in feet so eight feet four feet pretty pretty daunting problem here kind of one of the ones that if knew better about what was provided on it could have skipped it but if you're watching the video you know it's probably something that you want to try don't think i'm going to sign it to my students just for this sake i'm going to do both of these subjects in one and there's no way in heck i'm going to make them do you know five hours worth of homework or whatever this is going to end up being here all right so at least on there on their own pacing you know what mean okay so let's find the volumes of each of these things let's find the volume of the cone we know the height is eight we know its radius is six so the area of the base right 6. the b1 the area of the base is pi squared so that's 36 pi square feet so the volume of the cone is one-third area of base times height third of 36 is 12 1270 is 96 so this is 96 pi so if we take 96 pi we get that you know what i'm going to leave is 96 pi because got to subtract later so let's see volume 1 is 96 pi cubic feet so there's my start there 96 pi cubic feet i'm going to take away the volume of the cylinder so let's find that volume of that cylinder now the area of its base because its radius of the base is 3 feet that's established right there know it's very confusing looking drawing now but the area of its base is three feet so excuse me the the radius is three feet so the area of its base is pi times three squared which is nine pi square feet so the volume the total volume of that one is going to be pi times excuse me volume volume the volume of this was to say pi squared again 1 3 area of base times what was the height four it's right there times four times height third of nine is three three times four is twelve so that'll give us 12 pi cubic feet right there so the volume of the total have to subtract 12 pi from 96 pi volume of total equals 96 pi minus 12 pi which is 84 pi cubic feet that is an exact answer rounded we are going to get keep boxing boxed more on this problem than ever have before and should only be boxing the final thing so 84 times pi here we're gonna get 263.89 so 263 0.9 cubic feet there we go all right beautiful is that it one more question all right so word problem the roof of grain silo is in the shape of cone so this top part here that you can see we have cone on the roof it's kind of it's not really 3d i'm just drawing the triangle kind of thing the inside radius is 20 feet and the roof is 10 feet tall so okay might might end up redrawing all this right here so got cone right here got cone right here with boy i'm sorry about that drawing it's easier to do like on whiteboard or piece of paper so you guys should be fine with radius of 20 and height of 10 all right below the cone is cylinder 30 feet tall and that means it has the same radius but the height of the cylinder going down is 30 feet tall what my assumption is they could fill up they could fill up the grain silo the entire way otherwise they'd probably be asking about that volume so that height there is 30 feet what is the volume of the silo so yeah think the whole thing here so we're gonna add of course one with the other call this one figure one and figure two so the volume of the cone well the area the base is going to be the same for both of them pi squared which is 400 pi square feet the height of the cone is 10 feet so the total volume of the cone is one-third area of base times height neither of these can divide by three so i'm going to leave this as four thousand pi over three cubic feet for now the cubic feet isn't down with the three it's just couldn't fit it so there's the volume of that thing i'll leave it as that right now before do any you know no rounding early or anything because dividing by three would leave some repeating decimals if you think about that you know what i'm letting that giants game kind of flow in the background without me seeing it so i'm going to rewind before get spoiled on anything so sorry if you're going to re-watch parts again want to make sure to watch some of that later the figure two here has the same area of base right but this height instead is 30 feet and we're not going to take third of it but the volume of the cylinder there is area of base 400 pi times height of 30. so that's gonna be 12 000 guess 12 000 pi cubic feet and we are adding one with the other okay so here we go now by the way four thousand pi over three it's kind of close it's little more than four thousand because pi is three point one four so it's something that's little close to four thousand with that you know okay so the volume of the total there really would like to check all these answers to be sure did they ask us to they didn't ask us to round don't want it this this exact answer doesn't really work out this time anymore because the third part but will will round it i'll round to the nearest tenth because that's what they've been doing this whole time it's not perfect but will give kind of two different versions of an answer here so i'm going to add those two things there you know as an exact answer here let me give an exact one because this would be 36 000 pi over 3 the twelve thousand is common denominator plus four thousand pi over three so that's going to give you forty thousand pi over three cubic feet that's an exact you can't see that that's an exact answer right there and then an approximated answer forty thousand pi over three it's gonna be little more than forty thousand right so forty one thousand eight hundred eighty seven nine 41 887.9 cubic feet that's how much grain you could fit in there my gosh there we go all right there's part is that part okay part if one cubic foot of wheat is approximately 48 pounds and the farmer's crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo all right so this is like proportionality if you will if there are 48 pounds per every one cubic foot then and now we're going to compare to the 2 million but the question is how many pounds are there in the silo if we have this many cubic feet to work with personally we should be using an exact answer but don't think they're going to get so close to 2 million like within like single pound that it's going to make the difference so i'm just going to go ahead and use the to write on here still have the exact answer in my calculator right here to work with so am going to be calculating with this one but i'm just going to be writing this out out loud on paper you cross multiply 1 times is so we're going to figure out what this value is and got to do this number times 48. so times 48 you can't see the numbers but it's 0.298 so i'm getting as about and i'm just again i'm going to round this one because now we have our number so 2010619.298 so 0.3 that's pounds so going back here that cross multiplication allows us to get that answer will all the wheat fit in the silo is this million so that's two million ten thousand six hundred nineteen point three pounds will two million pounds of wheat fit in silo that is capable of holding two million plus pounds yes because this is more than two million this is how much wheat is going to go into what it can hold yes is that answer and math allowed us to decide that one there and that is the last question there and finished right before the hour so that ought to do it for this one guys is mr robinson yes did realize that this was going to be little longer because of all the drill and kill nature of this stuff sure hope that it was still fun for you to kind of play with and you know do some things with so let me know if you have any questions in the comment section down below let me know how many answers may or may not have watched really could have watched lot of them know that caught some other ones which is great i'm always glad to do those because made an error in the previous one and listen can't change that video anymore i'm not going to re-record if you know what mean and already deleted the video from my thing so you know that's just me and what do so thank you guys so much for watching thank you for being here within with me for an hour now it's passed take care go giants