Question

Find the dimensions of the rectangle with area 225 square inches that has a minimum perimeter, and then find the minimum perimeter.

Answer 1

minimum perimeter is achieved when the rectangle is a square.

Answer 2

since 225 is 15^2 you could guess but anyway:
Lw = 225 so L = 225/w
p = 2 L + 2 w
p = 2 * 225/w + 2 w
dp/dw = 0 at min = 2 [ -225 ] / w^2 + 2
225 = w^2
15 = w

Answer 3

the rectangle with max area per perimeter is a square

L * W = 225 ... L = 225 / W

p = 2 (L + W) = 2 [(225 / W) + W] = (450 / W) + 2 W

dp/dW = (-450 / W^2) + 2 ... the 1st derivative is zero at the minimum p

(-450 / W^2) + 2 = 0 ... (-450 / W^2) = - 2 ... 225 = W^2 ... W = 15