Source: Extra Practice
Among all rectangles with a fixed perimeter P, which shape has the maximum area?
Options
Option A
A rectangle with length twice the width
Option B is correct
A square
Option C
A rectangle with width very close to zero
Option D
Area is constant for all rectangles of same perimeter
Explanation
Let sides be x and y. Perimeter P = 2(x+y), so y = (P/2) - x. Area A = xy = x(P/2 - x) = (Px/2) - x². To maximize area, dA/dx = P/2 - 2x = 0, which gives x = P/4. Since x = P/4, then y = P/2 - P/4 = P/4. Because x = y, the rectangle is a square.