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

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. This should be clear and easy for a reader to follow (even if the reader may not be well-versed in mathematics). This means you should justify your steps in solving the problem.

Answer 1

minimum surface area = a cube's surface area with each side = cube root of 10,000 = 10 inches

minimum surface = 6 times the area of one side = 6 x 10^2 = 6x100 = 600 square inches

which at 1 cent per square inch = Cost of $6

if bottom cost twice as much as sides and top

then

x=side = about 11.856 inches = bottom's sides

with h= about 10000/11.856^2 = 71.138 inches height

but

2 cents per in^2 re-enforced bottom with 1 cent per in^2 for 4 sides and for the top

then minimum cost will when MC = Marginal Cost =0 with TC = Total Cost

= C(x)= .01(4)(hx) .01x^2+.02x^2

where x = side of the square bottom and top,

and h= height of the rectangular prism = Volume divided by bottom area = , 10000/x^2

C(x) = .05(10000/x^2)(x) + .03x^2= .05(10000/x) + .03x^2

= 500/x + .03x^2

C'(11.856) = 500/11.856 + .03(11.856^2)

= 42.173+4.217= $46.39 = minimum Cost

x = about 11.856 inches = sides of the bottom

h = about 71.138 inches = height

re-enforced bottom at twice the cost compared to the sides

costs an extra $40.39 per container

but you want an algebra step by step solution, not calculus derivative,

so this isn't the steps you want,

but if you do your algebra solution correctly you should get the same

costs $6, $46.39 and $40.39 difference

so the above is useful whether you understand it or not, if you don't know calculus