<PRE><B>Question 61442</B>
A rectangular box is to have a square base and a volume of 20 feet cubed. If the material for the base costs $.30/square foot, the material for the sides costs $.10/square foot, and the material for the top costs $.20/square foot, determine the dimensions of the box that can be constructed at minimum cost
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Let y = the cost of the box
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Let x = one side of the square base
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Let h = height of the box
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We know that the box has to be 20 cu ft:
x^2*h = 20
h = (20/x^2)
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Costs:
Base = .30(x^2)
Top = .20(x^2)
1 side = .10(x*(20/x^2)) = .10(20/x)
4 sides = .40(20/x) = 8/x 
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Total cost:
y =  Base + Top + 4 sides 
y =.30(x^2) + .20(x^2) + 8/x
y = .5x^2 + 8/x
Graph it:
{{{ graph( 300, 200, -6, 6, -6, 12, .5x^2 + (8/x)) }}}
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Looks like min cost (y) occurs when x = 2 ft, looks like a cost(y) of $6.00
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Then the height = 20/2^2 = 5 ft
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What is the cost then?
top .30(2^2) = 1.20
bot .20(2^2) = 0.80
hth .10(2*5) = 1.00 * 4 sides = $4.00
Total cost = $6.00 like the graph indicates.
 
:
Check: 2 * 2 * 5 = 20 cu ft as required

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