A rancher has 1000 feet of fencing with which to enclose two adjacent rectangular corrals with a interior partition (consider as one pen).

Use calculus to determine what the external dimensions of the pen that will maximize the enclosed area.
Write your answer in the form length by width. No decimals. Enter fractional answers in the form a/b

I don't understand the geometry of the Interior pen.

It basically looks like a rectangle with a line drawn right through the middle of it. So it looks like there are two pens connected as one.

25 feet by 30 feet

Ada

To find the external dimensions of the pen that will maximize the enclosed area, we can start by setting up an equation expressing the perimeter of the pen in terms of its dimensions.

Let's denote the length of the pen as L and the width as W. Since we have two adjacent rectangular corrals with an interior partition, we need to consider the length of fencing required for the perimeter of both corrals and the partition.

The perimeter of the two corrals is given by 2L + 2W. Since there is a partition that splits the pen into two corrals, we need to add an extra length of fencing equal to L for the partition. Therefore, the total length of fencing required is 2L + 2W + L = 3L + 2W.

According to the problem, the rancher has 1000 feet of fencing available, so we can express this constraint as:

3L + 2W = 1000 (equation 1)

Now, we need to express the area of the pen in terms of its dimensions. The area of a rectangle is given by A = L * W.

To maximize the enclosed area, we need to maximize the function A = L * W.

Next, we can express one variable in terms of the other using equation 1:

3L = 1000 - 2W
L = (1000 - 2W) / 3

Substitute this value of L in terms of W into the area equation:

A = [(1000 - 2W) / 3] * W

To find the maximum value of A, we can take the derivative with respect to W and set it equal to zero to find the critical points:

dA/dW = (1000 - 4W) / 3 = 0

Solving this equation, we find:

1000 - 4W = 0
4W = 1000
W = 250

Now, we can substitute this value of W back into equation 1 to find L:

3L + 2(250) = 1000
3L + 500 = 1000
3L = 500
L = 500 / 3

Therefore, the external dimensions of the pen that will maximize the enclosed area are:

Length L = 500 / 3
Width W = 250

So, the answer is length by width: 500/3 by 250.