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Question

A rectangular box is to have a square base and a volume of 192 cubic feet. If the material for the base costs 20 cent/square foot, the material for the sides costs 10 cent/square foot, and the material for the top costs 40 cent/square foot, determine the dimensions of the box that can be constructed at minimum cost.

Answer 1

Hi,

Let x = length of a side of the square base. Then its area is x^2. The area of the top is also x^2.

To find the volume of a rectangular box, you multiply length x width x height, so V = l x w x h. Since we don't know the height, we'll call it h for now. We know its volume is 192 cubic feet, so V = 192. Filling in what we have, our volume formula becomes:

192 = x * x * h
192 = x^2*h

Solving this for h, we get h = 192/x^2

Since each side is a rectangle that has a base of x and a height of h or 192/x^2 (<== we'll use that one), the area of one side is x * 192/x^2, which simplifies to 192/x. Since there are 4 of these sides their total area is 4*192/x or 768/x.

So area of base = x^2
area of 4 sides = 768/x
area of top square = x^2

Since the material for the base costs 20 cent/square foot, the material for the sides costs 10 cent/square foot, and the material for the top costs 40 cent/square foot, the price for these would be the above areas in square feet times these per square foot prices. Their costs are:

cost of base = 20x^2
cost of 4 sides = 10 * 768/x = 7680/x
cost of top square = 40x^2

All of these costs are in cents, but we can find the minimum cost whether they're all in dollars or in cents.

Total cost woul be the sum of these, so total cost is found by:

y = 20x^2 +7680/x + 40x^2

This simplifies to

y = 60x^2 +7680/x

If you put this equation into your calculator, we want to look for the graph's lowest point, because that would be where the minimum cost would occur. You will need to adjust your window to xmin=0, xmax = 20, ymin = 0, ymax= 6000, yscl = 1000 to see your graph easily. Using the minimum command you will find when x = 4 and its minimum cost Y is 2,880 cents or $28.80. This occurs at (4, 2880).

Dimensions of the box when x = 4 are 4 by 4 by 12. This gives the area of 192.

I hope this helps. :-)

Answer 2

Volume =H L^2= 192.
Cost = 20L^2 + 10 x 4 x HL + 40L^2
1 bottom, 4 sides and 1 top
Rewrite the cost function in terms of either H or L using the volume relation to relate L and H.
Take first derivitive of cost function.
Set that derivititive to zero.
Solve for the dimension used in the cost function
If necessary, discard negative or imaginery results.

Answer 3

let the base be b*b and the height h
volume = b*b*h = 192
volume is constant so 2bb'h + b^2h' = 0
so 2b'h + bh' = 0
the material is essentially the area of top sides and bottom
a = 20*b^2 + 10*4*b*h + 40*b*b
= 60 b^2 + 40 bh
a' = 120bb' +40b'h+40bh'
= 120bb'+40b'h-80b'h
= 40b'( 3b-h)
for minimum h = 3b
so 3b^3 = 192 so b^3 = 64 = 4^3
dimensions are 4 ft by 4 ft by 12 ft