MATH HELP: United Parcel Service has contracted you to design a closed box with a square base that has a volume of 10,000 cubic inches

Question

United Parcel Service has contracted you to design a closed box with a square base that has a volume of 10,000 cubic inches. Initially, they would like to know the dimensions of such a box with minimum surface area. 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 compare to the dimensions of the reinforced box which costs the least?

Provide:

A short discussion on your plan for solving the problem.

A step by step solution of the problem.

Answer 1

Least material would be a cubic box, with all sides equal

The volume = 10,000 cubic inches = 10^4 in^3

the sides are the cube root of the volume, cube root of 10,000 cubic inches = 10x(10)^1/3 inches 21.54 inches or almost 2/3 of yard for each side

When the base becomes twice as expensive, that will make the base smaller and the other sides longer, to minimize cost

Let b=the length of one side of the base, which is bxb

Let h= the length of the height of the box

Then the cost is C=2(bxb) + 4(hxb) + bxb. bxb is the area of the top & bottom, hxb is the area of a side

C=3b^2 + 4hb the volume = 10,000=area of base times height = hb^2 h=10000/b^2 substitute that expression for h into the cost function

C=3b^2 +4(10000/b) =3b^2 + 40000b^-1

To minimize cost take the derivative, C' = 0, and solve for b

C' = 6b-40000b^-2=0 Multiply by b^2

6b^3 - 40000 0

b^3 = 40000/6 = 6666 2/3

b = cube root of 6666 2/3 = 18.82 inches for the sides of the base, a little smaller than 21.54 The base shrinks a little and the sides will be higher to compensate for the smaller base to get the same volume

h=10000/b^2 = 10000/18.82^2=28.23 inches

Surface minimizing dimensions are 21.54 x 21.54 x 21.54 a cube with each side 10 times the cube root of 10.

Cost minimizing dimensions are 28.23 x 18.82 x 18.82

both have volume of 10,000 cubic inches. due to rounding approximations, the decimals are slightly less than required, but rounded to 2 decimals.

Surface area is minimized by a cube for any rectangular box. A sphere would be less surface area but that's not a box.

Surface area = 2b^2 +4bh h=volume divided by base area = 10000/b^2

substitute that expression for h, then take the derivative, set = 0, solve for b

A=2b^2+4(10000/b)

A'=4b -40000b^-2 = 0

Divide by 4

multiply by b^2

b^3 - 10000=0

b= cube root of 10000 = 10(10)^1/3 or 21.54

h=40000/b^3 = 40000/21.54^3 = 21.54

surface minimizing area box is 21.54 x 21.54 x 21.54 or a cube with each side the cube root of the volume