A box is constructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $2 per square foot

Question

A box is constructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $2 per square foot and the metal for the sides costs $7 per square foot. Find the dimensions that minimize cost if the box has a volume of 50 cubic feet.

Length of base x=_____

Height of side z=______

Answer 1

Let L = Length

Let L = Width ( property of squares)

Let H = Height

Then the volume V = L*L*H = 50ft^3

The cost for the top and bottom is ($2/ft^2)(L*Lft^2 + L*Lf t^2) = ($2/ft^2)(2L^2ft^2)= $4L^2

The cost of the sides = 4*$7*L*H= $28L*H

So you want to minimize the cost c = $4L^2+ $28L*H subject to L*L*H = 50ft^3

Just get rid of one variable in the c equation, take the derivative of c and ....