00:01
We have a cylinder that has an open end, and we're told it has a radius and a height, and we're also told that the volume is 32.
00:09
So i'm going to say the volume is 32, which comes from pi r squared times the height.
00:15
I'm going to go ahead and solve for the height, just because i know i'm going to need to eventually.
00:19
So the height is 32 divided by pi r squared.
00:23
We want to minimize the surface area.
00:26
So the area is going to come from pi r squared, which is the base, plus, the surface area of the side of the cylinder, which is going to be the circumference of the circle, so 2 pi r times the height.
00:43
Since we know the equation for the height, we can plug that in and get the area in terms of just the radius.
00:49
So we get pi r squared plus 2 pi r times this 32 over pi r squared.
00:59
And things will start to cancel.
01:02
So r will cancel here.
01:04
So i get finally that the area is equal to pi r squared plus two times 32 is 64 and i forgot to mention the pi is canceled earlier and then there's just an r to the negative one i wrote it that way just because i know i need to take a derivative next and that's easier so let's do that let's take a prime is equal to 2 pi r because that's the derivative of pi r squared minus 64 r to the negative 2 well r equals 0 doesn't make any sense...