A box with a square base and open top must have a volume of 108000 cm³. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area...

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Extracted text: A box with a square base and open top must have a volume of 108000 cm³. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of x.] Simplify your formula as much as possible. A(x) : Next, find the derivative, A' (x). A"(x) = %3D Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by x] A' (x) = 0 when x = We next have to make sure that this value of x gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x). A"(x) Evaluate A"(x) at the r-value you gave above.