SOLUTION: A closed box with square base is to be built to house an ant colony. The bottom of the box and all four sides are to be made of material costing $2/ft2, and the top is to be constr

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Question 1075632: A closed box with square base is to be built to house an ant colony. The bottom of the box and all four sides are to be made of material costing $2/ft2, and the top is to be constructed of glass costing $4/ft2. What are the dimensions of the box of greatest volume that can be constructed for $72? (Enter the dimensions (in feet) as a comma separated list.)

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let each of the 2 base dimensions = +x+
Let the height dimension = +y+
The area of the 4 sides = +4%2Ax%2Ay+
The area of the base = +x%5E2+
The area of the top is also +x%5E2+
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+2%2A%28+x%5E2+%2B+4%2Ax%2Ay+%29+%2B+4%2Ax%5E2+=+72+
+2x%5E2+%2B+8%2Ax%2Ay+%2B+4x%5E2+=+72+
+8%2Ax%2Ay++=+72+-+6x%5E2+
+y+=+9%2Fx+-+%283%2F4%29%2Ax+
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The volume is:
+V+=+x%5E2%2A%28+9%2Fx+-+%283%2F4%29%2Ax+%29+
+V+=+9x+-+%283%2F4%29%2Ax%5E3+
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If volume +V+ is maximized, then the cost of
construction will still be $72 total.
I will assume that this is a calculus problem, and
I will set the slope of +V+ equal to zero.
slope = +9+-+%289%2F4%29%2Ax%5E2+=+0+
+%289%2F4%29%2Ax%5E2+=+9+
+x%5E2+=+4+
+x+=+2+
and
+y+=+9%2Fx+-+%283%2F4%29%2Ax+
+y+=+9%2F2+-+%283%2F4%29%2A2+
+y+=+4.5+-+1.5+
+y+=+3+
The dimensions in feet are:
2,2,3
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check answer:
Here's the plot of volume, +V+, on vertical axis
and base dimension, +x+, on the horizontal axis.
+graph%28+400%2C+400%2C+-5%2C+5%2C+-20%2C+20%2C+9x+-+%283%2F4%29%2Ax%5E3++%29+
It looks like +V+ is maximized when +x+=+2+
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Get a 2nd opinion if needed