00:02
So in this question, they say that a closed box with a square base is to have a volume of 1 ,500 cubic inches.
00:11
So we are going to have a closed box with a square base, and it's going to have a volume of 1 ,500 cubic inches.
00:24
They say that the material for the top and bottom of the box costs 1 .5 cents per square inch.
00:30
The material for the sides of the box costs 2 cents per square inch.
00:36
I'm going to let x denote the length of a side of the square base and let h denote the height of the box.
00:48
Now, in part a, i want to write down a function for the cost of the material for the box in terms of x.
00:58
Now, what do i know? i know first of all that my volume was 1500.
01:08
That means that x squared times h is 1500 this time.
01:17
What do i know about my cost? well, first, i have the cost of the bottom.
01:26
The bottom has an area of x squared, and it costs 1 .5 cents per square inch.
01:33
So the bottom costs 1 .5 cents per square inch.
01:38
X squared.
01:40
Similarly, the top is going to cost 1 .5x squared as well.
01:48
Now i have my four sides.
01:52
Now, each of the four sides has an area of x times h.
01:58
And so in total, i have an area of 4xh there.
02:03
The material for the sides costs two cents per square inch.
02:07
So i'm going to multiply by two.
02:10
Simplifying.
02:12
My cost is equal to 3x squared plus 8xh.
02:20
Now, if x squared times h is equal to 1 ,500, this implies that my h is equal to 1 ,500 over x squared.
02:31
I'm going to take that expression for h, and i'm going to plug it in, so that my cost is 3x squared plus 8x times 500 over x squared.
02:48
Simplifying, my cost is 3x squared, plus 8 times 1 ,500, that's 12 ,000, x over x squared gives me 12 ,000 over x.
03:00
There's my answer to part a...