00:01
So in this question we have a salinical package that's going to be sent by a postal service.
00:08
So a cylindrical package is going to be sent by a postal service, and you can have a maximum combined length and girth, which is the perimeter of a cross section of 114 inches.
00:23
I want to find the dimensions of the package of maximum volume that can be sent.
00:29
Our cross sections are circular.
00:32
So here i have my height, and then here is going to be the circular cross section.
00:40
They said the length plus the girth is going to be 114 inches this time.
00:48
My length, well, that's my height here, and my girth, that's the distance around this circular cross section.
00:57
And so that's called the circumference of the circle.
01:01
That's 2 pi r.
01:03
And so my height plus my 2 pi r is equal to 114 this time.
01:09
And if that's the case, my goal is try and maximize the volume of this cylinder.
01:17
Well, the volume of a cylinder is pi r squared times the height.
01:23
Currently, that volume is a function of two variables.
01:27
And i want to squash that down to the world of one barrier.
01:31
To do that, i'm going to go to my constraint and solve for one of my variables.
01:36
It's easiest here to solve for h.
01:39
My h is equal to 114 minus 2 pir.
01:46
I'm going to take that and plug it into my volume for you.
01:49
So that my volume is pi r squared times the quantity of 114 minus 2 pir.
01:59
I'm going to distribute that pi r squared so that my volume is 114 pi r squared minus 2 pi squared r cubed.
02:13
To maximize this volume, i'm going to need my derivative.
02:17
I need my v prime.
02:20
That'll be equal to 228 pi times r minus 6 pi squared times r squared...