Among all rectangles that have a perimeter of 88, find the dimensions of the one whose area is largest. Write your answers as fractions reduced to lowest terms.

Answer 1

P = 88 = 2(L+W)

44 = L+W

rectangle with largest area is a square with L=W

44 = L+L = 2L

L = 44/2 = 22 = W

dimensions are 22 by 22

A = LW = 22(22) = 484 = largest area for a rectangle with perimeter = 88

if you don't realize a square is the rectangle with largest area for a fixed perimeter, you can use calculus to find the dimensions:

L+W = 44

W = 44-L

A= LW = (44-L)L = 44L -L^2

take the derivative and set equal to zero, solve for L

A'(L) = 44-2L = 0

2L = 44

L=44/2 = 22 long

W = 44-L= 44-22 =22 wide

dimensions are 22 by 22