Question

A rancher plans to make 4 identical and adjacent rectangular pens against a barn (see figure below), each with an area of 100 m^2. What are the dimensions of each pen that minimize the amount of fence that must be used?​

Answer 1

Answer:

10 by 10 meters

Step-by-step explanation:

In calculus terms, you can use the formula for the perimeter which is 2l + 2w, and the area l × w.

Then since the area is 100

l × w = 100. So w = 100/l

Then substitute this into the perimeter formula.

w = 100/l → 2l + 2(w) = 2l + 2(100/l) = 2l + 200/l.

Then take the derivative d/dx [take the derivative of a quantity with respect to x], by using a simpler method known as the power rule:

d/dx[x^n] = nx^n-1

so p = 2l + 200/l becomes p' = 2 - 200/l^2

Since the derivative of a constant is 0, to solve for l, set p' to 0

0 = 2 - 200/l^2 → -200/l^2 = -2 → 200/l^2 = 2 → 2l^2 = 200 → l^2 = 100 → l = √100, l = 10.

Now that we know the length is 10, We can substitute the length in the original equation to find the width.

l = 10 → w = 100/l → w = 100/10, w = 10.

Therefore the perimeter of each rectangular pen is 40, with a length of 10 and width of 10.

If you want to do this algebraicly, there are more steps involved