00:01
So in this question, we have a rectangular plot of land that is to be fenced in using two kinds of fencing.
00:10
Two of the opposite sides will say these two opposite sides, which we'll call x, will use heavy -duty fencing selling for $3 per square foot, while the two remaining sides will use standard fencing for $2 per foot.
00:29
So the other two sides, which we'll call why, they use standard fencing for $2 per foot.
00:38
And we ask, what are the dimensions of the rectangular plot of greatest area that can be fenced in for a total of $6 ,000? so what is the cost of constructing this fence going to be? well, on the left and right, it's $3 per foot.
01:03
So on the left, if i have x feet and it costs $3 per foot, the cost of constructing that left side is 3x.
01:15
Similarly, the cost of constructing the right side of this rectangle is 3x.
01:22
The cost of constructing the top of this rectangle is going to, to be 2y because it's $2 per foot and there's y feet.
01:34
So 2y for the top and similarly 2y for the bottom.
01:42
So that's the cost of constructing this rectangle and that's supposed to be equal to $6 ,000.
01:49
Now simplifying a little bit we've got 6x plus 4y equals 6 ,000.
01:59
Or dividing by 2 to simplify things a little bit, we have 3x plus 2y equals 3 ,000.
02:13
So we know this to be true.
02:17
And what are we trying to do here? we are trying to maximize the area of this rectangle.
02:24
I'm trying to maximize my area, which is x times y.
02:30
Now, i need to squash this area function, the thing i'm trying to maximize, down from the world of two variables to the world of one variable.
02:43
How do i do that? well, i'm going to go to my constraint, and i'm going to solve for one of the variables.
02:50
Specifically, i'll solve for why.
02:52
It doesn't matter which one you solve for, but i'll solve for why.
02:56
It's a little nicer.
02:57
So, let's see, i know that 2y is equal to 3 ,000 minus 3x...