What is the minimum possible perimeter for a rectangle whose area is 100 square inches?

Question

How to solve the question in the title?

Answer 1

Let L and W represent the length and width, respectively, of the rectangle. The perimeter P = 2(L + W), and since the area is 100 in² we have 100 = LW, or W = 100/L. Therefore P = 2(L + 100/L). Completing the square,

P = 2[L - 20 + 100/L + 20] = 2[(√L - 10/√L)² + 20] = 2(√L - 10/√L)² + 40

So the perimeter P is at least 40 in, and equals 40 in exactly when √L = 10/√L, i.e., L = 10 in. So the minimum possible perimeter is 40 in (when the rectangle is a square).