00:01
Consider this figure representing the rancher, and we are given that the rancher has 200 feet of fencing.
00:09
Now we want to find the dimensions x and y that will maximize the area.
00:15
So from this, we know that the objective function is the area function.
00:20
That's a, which is equal to length times width, that'll be y times 2x.
00:31
Or that's the same as 2xy.
00:35
Next you want to write our objective function in terms of one variable.
00:40
Let's say you want to write this in terms of x.
00:42
So we will use the fact that the total amount of fencing is 200 feet.
00:51
And if you look at this figure, our 200 feet, this is equal to 4x plus 3y.
01:01
So if i'm going to solve for y in terms of x, you have y equal to 200 minus 4x all over 3.
01:11
So then our a now will be equal to 2x times 200 minus 4x over 3.
01:22
Or that's the same as 8 over 3 times 50x minus x squared.
01:34
And then since you already have our objective function, you will then find the derivative of this.
01:40
So differentiating, we have a prime of x, that's equal to 8 over 3 times 50 minus 2x.
01:50
And then you will set this to 0 to find the critical number.
01:55
Setting this to 0, we have 8 over 3 times 50 minus 2x equals 0.
02:02
We get 50 minus 2x equals 0, or that's going to be negative 2x equals negative 50, or that's x equal to 25.
02:13
Note that from here, we can tell that our x is restricted because this numerator has to be greater than or equal to 0, which means that our x will be less than or equal to 200 over 4, or that's about 50...