00:01
So in this question, they say that an open top cylindrical container is to have volume 2744 cubic centimeters.
00:10
I want to know what dimensions, radius, and height will minimize that surface area.
00:17
So here i have a cylindrical container.
00:23
I have a cylindrical container, and it's going to have a volume of 2744 cubic centimeters.
00:30
Now, the volume of a cylinder is pi r squared times my height.
00:38
So pi r squared times my height this time is 2744.
00:44
Now, i am trying to minimize my surface area.
00:52
So my surface area has a couple of components.
00:55
First, i have the bottom.
00:57
The area of that bottom is pi r squared.
01:02
Then i have the area of the curved side.
01:08
The area of that curved side called the lateral area of a cylinder is 2 pi rh.
01:15
So there's my surface area that i'm trying to minimize.
01:19
And i am trying to do so subject to the constraint that pi r squared h equals 2744.
01:28
Notice there is no top, so there's not a second pi r squared.
01:32
This time.
01:34
I'm going to start by squashing that surface area formula down from the world of two variables to the world of one variable.
01:44
So if pi r squared h is equal to 2744, that tells me that my h is equal to 2744 over pi r squared.
01:58
I'm going to take that expression for h and plug it in to my surface area formula.
02:06
My surface area is pi r squared plus 2 pi r times 2744 over pi r squared.
02:19
And so let's simplify...