Quick help :United Parcel Service

Question

Problem United Parcel Service has contracted you to design a closed box with a square base that has a volume of 10,000 cubic inches. (a) Initially, they would like to know the dimensions of such a box with minimum surface area. (b) They are considering creating the box with a reinforced bottom. The box would be made from a cardboard material costing 1 cent per square inch to build the walls and top, however the bottom would be constructed with stronger material, costing 2 cents per square inch to build. How do the dimensions of the box having the minimum surface area [as in case (a)] compare to the dimensions of the reinforced box, which costs the least [as in case (b)]?

(No calculus derivatives) 

A step-by-step solution of the problem.

Answer 1

A) Let's assume that the dimensions of the square base are x by x, and the height of the box is h. Then, the volume of the box can be expressed as:

V = x²h = 10,000

Solving for h, we get:

h = 10,000 / x²

The surface area of the box can be expressed as the sum of the area of the base and the area of the four sides:

S = x² + 4xh

Substituting the expression for h from above, we get:

S = x² + 4x(10,000 / x²)

Simplifying, we get:

S = x² + 40,000 / x

To find the dimensions of the box with minimum surface area, we need to find the value of x that minimizes S. We can do this by finding the critical points of S, which are the values of x where the derivative of S with respect to x is equal to zero:

dS/dx = 2x - 40,000 / x²

Setting dS/dx equal to zero and solving for x, we get:

2x - 40,000 / x² = 0

Multiplying both sides by x² and rearranging, we get:

2x³ = 40,000

x³ = 20,000

Taking the cube root of both sides, we get:

x = 26.23

Therefore, the dimensions of the box with minimum surface area are approximately 26.23 inches by 26.23 inches by 14.38 inches (height).