A Farmer is making a pen to contain his cows. What is largest size (by area) rectangular pen that he can make with 150 ft. of fencing?

Answer 1

150/4 = 75/2 = 37.5 feet for each side of a square

largest area = 37.5 squared = 1,406.25 square feet

a square has the largest area for a rectangle with given perimeter

but if you want to calculate it with calculus

P= 150 = 2L+2W, L+W = 75, L=75-W

Area = A = LW = (75-W)(W) = 75W -W^2 take the derivative and set = 0

A'(W) = 75 -2W = 0

2W = 75

W = 75/2 = 37.5 ft. = the width which maximizes area

L=75-37.5 = 37.5 ft = length which maximizes area

A=LW = 37.75(37.75) = 1,406.25 ft^2 = maximum area of a rectangle with perimeter = 150 ft.