Question

Levi wants to build a rectangular pen for his animals. One side of the pen will be against the barn; the other three sides will be enclosed with wire fencing. If Levi has 400 feet of fencing, what dimensions would maximize the area of the pen? (a) Let w be the length of the pen perpendicular to the barn. Write an equation to model the area of the pen in terms of w

Answer 1

Explanation:

Let's call the length of the pen perpendicular to the barn "w", and the length parallel to the barn "l".

The total amount of fencing available is 400 feet. The side against the barn will be of length l, and the other two sides will each be of length w. Therefore, the total length of fencing required is:

l + 2w

This length of fencing must equal 400 feet:

l + 2w = 400

We want to maximize the area of the pen. The area of a rectangle is given by:

A = lw

We can use the equation l + 2w = 400 to solve for l:

l = 400 - 2w

Substituting this expression for l into the equation for the area, we get:

A = w(400 - 2w)

Simplifying:

A = 400w - 2w^2

So the equation to model the area of the pen in terms of w is:

A = 400w - 2w^2