Question

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.)

Answer 1

• Base Length = 7.368cm
• Height = 0.737cm.

Explanation:

Volume of the jewelry box=
`40cm^3`

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

`l^2h=40\\h=(40)/(l^2)`

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

Since we have a square base

Total Surface Area =
`2(l\²+lh+lh)`

The Total Surface Area of the box
`=2l\²+4lh`

Nickel plating costs $1 per
`cm\³`

Silver Plating costs $10 per
`cm\³`

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

Therefore, Cost of the Material for the jewelry box
`=1(2l\²)+10(4lh)`

`Cost, C(l,h)=$(2l\²+40lh)`

Recall earlier that we derived:
`h=(40)/(l^2)`

Substituting into the formula for the Total Cost

`Cost, C(l)=2l\²+40l((40)/(l^2))\\=2l\²+(1600)/(l)\\C=(2l^3+1600)/(l)`

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

`C^(')=(4l^3-1600)/(l^2)`

`4l^3-1600=0\\4l^3=1600\\l^3=400\\l=\sqrt[3]{400}=7.368 cm`

Recall:

`h=(40)/(l^2)=(40)/(7.368^2)=0.737cm`

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