Question

David wants to build a rectangular fencing with the 5 identical parts for his animals. He has 780 feet of fencing to make it. What dimensions of each part will maximize the total enclosed area?

Answer 1

Explanation:

So we're looking at a rectangle split into 5 smaller rectangles. If the height of each rectangle is y and the width of each rectangle is x, then the amount of fencing is:

P = 6y + 10x

And the area of the large rectangle is:

A = 5xy

We know that P = 780:

780 = 6y + 10x

10x = 780 - 6y

5x = 390 - 3y

If we substitute this into our area equation:

A = (390 - 3y) y

A = -3y² + 390y

This is a vertical parabola pointing down, so we know the maximum is at the vertex, which is at -b/(2a). Or, we can use calculus to take the derivative and set to 0.

dA/dy = -6y + 390

0 = -6y + 390

y = 65

Solving for x:

5x = 390 - 3y

5x = 390 - 3(65)

5x = 195

x = 39

So each part will have a width of 39 feet and a height of 65 feet.