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.
