A rectangle has a perimeter of 20 in. Find the length and width of the rectangle under which the area is the largest.

Answer 1

P = perimeter, L = Length, W = width

P = 2(L + W) = 10

L + W = 5

W = 5 - L

Area = A = LW = L(5 - L) = $5L-L^2$

$\frac{dA}{dL}=5-2L$

At maximum 5 - 2L = 0; L = 2.5 and W = 2.5; that is with fixed perimeter the rectangle with the largest area is a square.