Question

A farmer will build a rectangular pen for some goats. A wall will form one side of the pen. The farmer has 36 m of fencing to form the other three sides. The farmer plans to build the pen so that it has its maximum possible area. What will be the dimensions of the farmer’s goat pen?

_____m by _____m

Answer 1

9m by 18m is the answer

Answer 2

The maximum area is obtained when half the fence (18 m) is used parallel to the wall. Then the remaining half of the fence forms the other two (equal, 9 m) sides.

The dimensions of the pen will be 9 m by 18 m.

_____

If x represents the width of the pen (two sides out from the wall), then the area is

... area = x·(36 -2x) = 2x(18 -x)

This expression describes a parabola that opens downward and has zeros at x=0 and x=18. The line of symmetry (and vertex) is halfway between these zeros, at x=9. Hence the width of the pen is 9 m and its length is 36-2·9 = 18 m.