2 Show that given a fixed perimeter C the rectangle with the

2) Show that given a fixed perimeter C, the rectangle with the largest area is a square. (Note: The formula for the area is the objective function. The formula for the perimeter gives the constraint.)

Intermediate Microeconomics with Calculus

Solution

Let the perimeter be C.

Length be x

Breadth be C/2 - x

Area, A = x (C/2 – x)

A = - x^2 + C/2 x

Derivative of A = - 2x + C/2

0 = -2x + C/2

2x = C/2

X = C/4

(C/2 – x) = C/2 – C/4 = C/4

Maximum area = x × (C/2 – x) = C/4 × C/4 = (C/4)^2

Answer: The maximum area is when length and breadth are same that is square.