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A box with a square base and open top must have a volume of 32000 cm3....

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Basic Math

A box with a square base and open top must have a volume of32000 cm3. We wish to find the dimensions of the box that minimizethe amount of material used.

Find the following:
1. First, find a formula for the surface area of the box in termsof only x, the length of one side of the square base. [Hint: usethe volume formula to express the height of the box in terms of x.]Simplify your formula as much as possible.

2. Next, find the derivative, A'(x).

3. Now, calculate when the derivative equals zero, that is, whenA'(x)=0.

4. We next have to make sure that this value of x gives aminimum value for the surface area. Let's use the second derivativetest. Find A\"(x).

5. Evaluate A\"(x) at the x-value you gaveabove.

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