what are the dimensions of a rectangular shape that will minimize while still giving 2500 square feet

Question

what are the dimensions of a rectangular shape that will minimize the amount of outside walls that they need while still giving them 2500 square feet of floor space

Answer 1

Let the shape of the rectangle have length L and width W.

 

Then area A = L * W = 2500

From this the width W = 2500/L

 

Assuming outside walls represent the perimeter of the rectangle.

 

Then P = 2(L + W) = 2(L +2500/L)

                            = (2/L)(L² + 2500)

                            = (2/L)(L² -100L + 2500) + (2/L)(100L)

                            = 200 + (2/L)(L - 50)²

 

The perimeter is a minimum when L = 50 ft, because any other value of L

makes the perimeter greater.

 

W = 2500/L = 50 ft

 

The dimension of the room should 50 ft x 50 ft