SOLUTION: A rectangular bos id to be made from two types of material. The top and btom cost $6.00 per square inch and the sides cost $2.00 per square inch. If the box is to hol 80,000 cubic

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Question 133745: A rectangular bos id to be made from two types of material. The top and btom cost $6.00 per square inch and the sides cost $2.00 per square inch. If the box is to hol 80,000 cubic inches and the top and bottom are squares, find the dimensions and the minumum cost.
Answer by ankor@dixie-net.com(22740) (Show Source):
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A rectangular box is to be made from two types of material. The top and bottom cost $6.00 per square inch and the sides cost $2.00 per square inch. If the box is to hold 80,000 cubic inches and the top and bottom are squares, find the dimensions and the minimum cost.
:
Let x = dimension, in inches, of the square top and bottom
:
Find the height in terms of x using the volume equation:
x^2 * h = 80000
h =
:
Surface area = 2(x^2) + 4(x*h)
:
Substitute for h in the above equation
SA = 2x^2 + 4(x*)
Cancel x
SA = 2x^2 + 4()
Write an equation for the surface area cost:
SA(cost) = 6(2x^2) + 2*4()
SA(cost) = 12x^2 + 8()
SA(cost) = 12x^2 +
graphing this equation, where y = the total cost of the material and x is the dimension of the square top and bottom

The value for x for minimum cost = 30 inches
Use h = to find the height
h =
h = 88.88 inches is the height
:
Find the cost:
Cost = 6(2*30^2) + 2(4*30*88.88)
Cost = 6(1800) + 2(10665.6)
Cost = 10800 + 21331
Cost = $32,131 per box which roughly agrees with our graph
:
We can say the box will be 30 by 30 by 89 inches and using these dimensions,
the cost will be about $32,160