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

Answer:

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

Step-by-step 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

$l^2h=40\\h=\frac{40}{l^2}$

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: $h=\frac{40}{l^2}$

Substituting into the formula for the Total Cost

$Cost, C(l)=2l\²+40l(\frac{40}{l^2})\\=2l\²+\frac{1600}{l}\\C=\frac{2l^3+1600}{l}$

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

$C^{'}=\frac{4l^3-1600}{l^2}$

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

Recall:

$h=\frac{40}{l^2}=\frac{40}{7.368^2}=0.737cm$

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