A jewelry box with a square base is to be built with silver plated sides, nickel plated bottom and top, and a volume of 40 cm3. If nickel plating costs $ 1 per cm2 and silver plating costs $ 10 per cm2, find the dimensions of the box to minimize the cost of the materials. (Use decimal notation. Give your answers to three decimal places.)

Step-by-step explanation:

Volume of the jewelry box=

The box has a square base and is to be built with silver plated sides and nickel plated top and base.

Therefore: Volume  = Square Base Area X Height = l²h

Total Surface Area of a Cuboid =2(lb+lh+bh)

Since we have a square base

Total Surface Area =

The Total Surface Area of the box

Nickel plating costs $1 per

Silver Plating costs $10 per

Since the sides are to be silver plated and the top and bottom nickel plated:

Therefore, Cost of the Material for the jewelry box

Recall earlier that we derived:

Substituting into the formula for the Total Cost

The minimum costs for the material occurs at the point where the derivative equals zero.

Recall:

The box which minimizes the cost of materials has a square base of side length 7.368cm and a height of 0.737cm.