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Calculus/volume/surface area

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Expert: - 6/2/2009


Question
If 1200 cm cubed of material is available to make a box with a square base and and open top, find the largest possible volume of the box. Use the following steps:
a. Find the volume in terms of x and h.
b. Find the surface area in terms of x and h.  Then solve for h in terms of x
c. Find the volume in terms of x.

Thanks for your time.

Answer
a. The base has width x, so the bottom area is x².
Multiply this by the height to get the volume.
The volume, then, is V = hx².

b. The surface area involves the area of the sides, each with area xh and the base x².  There are 4 sides, so the area is A = 4xh + x².

c. We know that hx² = 1200, so h = 1200/x².  Putting this into surface area gives us x² + 4x(1200/x²) = x² + 4800/x².  Setting this to 0 will git 2x^3 = 4800, so x^(1/3) = cube root(4800).  

Using hx² = 1200, you can put in x and find h.

From there, it is known that the volume V is hx².

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