Among all rectangles that have a perimeter of 180, find the dimensions of the one whose area is largest. Write your answers as fractions reduced to lowest terms.

Answer 1

180/4 = 45

45 by 45 is the rectangle with perimeter 180 that has the largest area

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

or if you want to use some calculus, take the derivative of the Area function, A, and set it = 0 Area = length times width.

P=180 = 2L + 2W

90 = L + W

L = 90-W

A= LW = (90-W)W = 90W -W^2

A'= 90-2W =0

W = 90/2 = 45 wide

L =90-W = 90-45 = 45 long

dimensions are 45 by 45