Question

A farmer has 336 feet of fencing to enclose 2 adjacent rectangular pig pens sharing a common side. What dimensions should be used for each pig pen so that the enclosed area will be a maximum? The two adjacent pens have the same dimensions.

Answer 1

x = 84 ft the largest side of the pens

y = 56 ft the common side is that length

Explanation:

Let call x the total length of one side, and y the other side, and the common side such as:

p(perimeter) = 336 = 2x + 3y y = ( 336 - 2x ) / 3

And the area of the whole area

A(t) = x * y A(x) = x* (336 - 2x)/3 A(x) = [336x - 2x² ]/3

Taking derivatives both sides of the equation

A´(x) =[ ( 336 - 4x )*3]/9 ⇒ A´(x) = 0 ( 336 - 4x )*3 = 0

336 - 4x = 0

x = 336/4

x = 84 ft and y = ( 336 - 2x ) / 3 y = ( 336 - 168)/3

y = 56 ft