Calculus Optimization?

Question

A rectangular field is to be bounded by a fence on three sides and by a river on the
fourth side. With 1000 meters of fencing at your disposal, what is the largest area you can
enclose, and what are its dimensions? Be sure to show that your answer gives the maximum
area! Please so work!

Answer 1

The area of the field is L W where L and W are the dimensions of the rectangle formed by the fence and the river.

Since you have 1000 feet of fence to play with,

2 L + W = 1000 so W = 1000 - 2L

Now the area is

A = L W = L ( 1000 - 2L ) = -2 L ^2 + 1000 L

To find the minimum or maximum, take the first derivative of A and set it equal to zero:

A' = - 4 L + 1000 = 0

Solve for L:

L = 250

So you form the maximum area with a rectangle of sides 250 by 500.