A shipping company must design a closed rectangular shipping crate with a square base. The volume is 1152ft3. The material for the top and sides costs $2 per square foot and the material for the bottom costs $7 per square foot. Find the dimensions of the crate that will minimize the total cost of material.

At minimum cost, Length = 8 foot, Height = 18 foot

Step-by-step explanation:

Volume of the closed rectangular shipping crate =

The crate has a square base from the given statement.

Therefore: Volume  = Square Base Area X Height =

=1152

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 Crate

= Area of Top + Area of Bottom + Area of Sides

The material for the top and sides costs $2 per square foot and the material for the bottom costs $7 per square foot.

Therefore, Cost of the Material for the Crate

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 dimensions that will minimize the cost of the box are: Length = 8 foot, Height = 18 foot

the given 6 = w/p is equivalent to 6p = w, where w is the total area of the walls and p is the area covered by a single can of paint.

knowing that 6 cans of paint will be needed, 6p = w becomes 6(6) = w = 36.   this tells us that the total area to be painted is 36 square units.