SOLUTION: Sally is building a box with a square base for her ant colony. The bottom of the box and all four sides are con- structed of glass costing $1/ft2, and the top is to be con- stru

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Question 236021: Sally is building a box with a square base for her ant
colony. The bottom of the box and all four sides are con-
structed of glass costing $1/ft2, and the top is to be con-
structed of glass costing $5/ft2. Sally has $72 to spend on
the box. How should she choose the dimensions so that the
volume of the box is greatest?
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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The bottom of the box and all four sides are constructed of glass costing $1/ft2,
and
the top is to be constructed of glass costing $5/ft2.
Sally has $72 to spend on the box.
How should she choose the dimensions so that the volume of the box is greatest?
:
Let x = the side of the square base and top
Let h = height of the box
5 of the sides cost $1 a sq/ft
Top cost 5x^2
:
We need to get h in terms of x
:
Top + bottom + 4 sides = 72
5x^2 + x^2 + 4(xh) = 72
6x^2 + 4(xh) = 72
4xh = 72 - 6x^2
h = %2872-6x%5E2%29%2F4x
Simplify
h = %2836-3x%5E2%29%2F%282x%29
:
Find the volume:
V = x^2*h
replace h
V = x^2*%28%2836-3x%5E2%29%29%2F%282x%29
Cancel x
V = x*%28%2836-3x%5E2%29%29%2F2
V = %28%2836x-3x%5E3%29%29%2F2
Write the equation as:
-1.5x^3 + 18x = 0
:
Graph this
+graph%28+300%2C+200%2C+-2%2C+5%2C+-10%2C+30%2C+-1.5x%5E3%2B18x%29+
Max volume when x=2
:
find h, when x=2
h = %2836-3%282%5E2%29%29%2F%282%2A2%29
h = %2836-12%29%2F%284%29
h = 24%2F4
h = 6
;
A box: 2 * 2 * 6 = 24 cu (as seen on the graph)
;
Check the cost;
Top + bottom + 4 sides = $72
5(4) + 4 + 4(2*6) =
20 + 4 + 48 = $72