A farmer intends to fence o a rectangular pen for his pig Wilbur, using the barn as one of the sides. If the enclosed area is to be 50 square feet, what is smallest amount of fence needed, in feet?
A)30
B)40
C)20
D)10
E)50
F)8
G) $20 \sqrt { 2 }$
H)45

Question

A farmer intends to fence o a rectangular pen for his pig Wilbur, using the barn as one of the sides. If the enclosed area is to be 50 square feet, what is smallest amount of fence needed, in feet?
A)30
B)40
C)20
D)10
E)50
F)8
G) $20 \sqrt { 2 }$ 
H)45

Quizplus · Accepted Answer

The minimum amount of fencing required to enclose a given area is achieved with a square shape. However, since one side uses the barn, the rectangle that minimizes the perimeter for a given area has two sides equal. For an area of 50 square feet, if one side (the side not using the barn) is x, then the other two sides (the ones needing fencing) are 50/x. The total length of fencing needed is x + 2(50/x). To minimize this, set the derivative equal to zero, or simply recognize that for a rectangle with a fixed area, the closer the sides are to being equal, the smaller the perimeter. Here, a 5 by 10 rectangle minimizes the fencing because it approaches the shape of a square, requiring 5 (barn side, not fenced) + 10 + 10 = 20 feet of fencing.

A Farmer Intends to Fence O a Rectangular Pen for His