Surface Area of a Cylinder 12 2

well let's look at the area of right cylinder and we're going to do this by taking circle down here and this plane will say this circle has radius of six feet and we're going to extrude it it's on the green blue plane and we're going to take this and extrude it into the blue axis to height of 15 feet so now we've got figure which has circle for base on the top and circle for the base on the bottom but now it's got this curved lateral area just like just like the label of soup can we can see we've got the two formulas on the left and the surface area is two times the base the bases are the two circles plus the circumference times the height the height means the height of the cylinder that's the 15 in this case we're going to split this one apart and and solve it with net just like we did with prism let's go do that now we'll work off the net view for this cylinder that we've just seen in 3d we know the two circular bases area of the circle is pi squared this circle has radius of six feet we know that the height of the cylinder is 15 feet and we know that the distance remember from this demonstration the distance that circle travels without skidding in one revolution is its circumference so its circumference times the height would be the area of this rectangle which would be the lateral area that curved surface which has been flattened out here just like the label on that campbell soup can so area circumference times the height circumference is pi so you could say it's pi or you could say it's 2 pi any of those three would give you the that green area so now let's clean this up little bit and write this expression down because the area of cylinder and this of course is right cylinder we've got the green expression plus two of the blue ones which are simply circles so it's two pi squared and let's substitute in for this particular figure and we have substituting in radius of and height of 15 and when we simplify we can see we have we generate two terms that can be combined we have 180 pi for the lateral area and 72 for the two circles they're still in terms of pi can combine them 252 pi square feet and then can pull out my handy dandy calculator and to the nearest tenth guess i'd come up with about 791 and 7 tenths square feet well anyone can work problem forward let's take this one backwards surface area and we're given surface area for this entire cylinder i've got two things to do straight away one is to factor out the two pi from both those terms and the other is to substitute this value this given value now did these two first because now can divide both sides of the equation by two pi so let's get that done and i'm going to take move this out of the way i've got let me see my memory is clear one zero nine that's divided by 2 divided by pi that's divided by 2 pi right there now i'm going to jot down those first five digits but the first thing i'm going to do i'm going to put this into the memory of my calculator so can come back to it all right that's good so now let me jot this down over here i'm going to show my division and say all right well guess i'm going to have to square and i've also done this substitution for the radius we have to square this 8.2 over here so let me pull go back to my calculator and that's in memory so i'm going to clear this right here and i'm going to take my 8 and 2 tenths i'm going to square it well that's 67 and 2400 square meters which can now subtract from both sides of the equation so minus and in the memory will be this expression so let's catch up i'm looking for that 107. let's catch up to that all right that's the squaring and that's the subtracting so i've got 107 and 3 500 square meters and have to divide that by 8.2 meters you notice we've carried units here i've subtracted square meters from square meters now i'm going to divide square meters by meters which of course will give me meters exciting so i'll take this and will divide by 8.2 and now i'm done let's see what wanted two decimal places so we wanted to round to the nearest hundredths thirteen and nine hundredths let's see what we've got there and there you go 13 and 900 finished well here's an interesting hypothetical question i'm going to take this red cylinder and i'm going to divide let me see both the height and its radius by radical five get the blue cylinder made them to scale for you but let's do the math and see how much smaller is the blue one than the red let's give it shot here let's start with this formula that we know surface area and i'm going to just do substitution i'm going to replace in each case the radius with the radius divided by radical 5 and the height with height divided by radical 5. now know we'd normally rationalize the denominator but we're not going to have to we're just going to square these look at this we've got square term here and we've got radical 5 times radical 5 in the denominator there so when simplify this i've got my new expression with my fives in the denominator and pretty straightforward here let's just factor out the fives from the denominator or another way of saying it factor out the one-fifth and you're there and you can see clearly this blue that means that this blue figure has surface area one-fifth of the red one or we could say that the red has surface area five times greater than the blue well here's an algebra exercise where we have given surface area for the cylinder and we're given relationship between the height and the radius the radius is twice the height so it's pretty squat cylinder that we're looking at here picture down in the lower right so i'm going to take our formula for surface area and let's do the substitution very conveniently this 108 pi square meters well contains factor of pi so i'm also going to factor out the 2 pi from the expression on the right and can divide both sides equation by 2 pi pretty straightforward now we're going to have to figure something out here let's replace radius with 2h since well since that's the ratio and now we can work it out like this we can we can substitute in here for the radius and notice when we square that be careful when you're squaring that's going to be 4h squared and of course i've got 2h times which is 2h squared moving on from here can combine these two terms 6h squared and over here i've just taken symmetric property and just reversed it like my variable on the left and now let me see divide both sides of the equation by 6 and then guess we'll take the principal or positive root and the height would then be three meters notice throw the units in here we've been keeping track of it we know what's the difference between square and linear measurements and height is certainly linear measurement and that would mean that the radius would be double that or six so we weren't asked that but there it is so you've got radius of six meters height of three meters and judging by the person inside that looks about right you