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A box with a square base and open top must have a volume of 97556 cm3cm3. 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 xx, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of xx.] Simplify your formula as much as possible. A(x)=A(x)=  

Next, find the derivative, A'(x)A′(x). A'(x)=A′(x)=  

Now, calculate when the derivative equals zero, that is, when A'(x)=0A′(x)=0. [Hint: multiply both sides by x2x2.] A'(x)=0A′(x)=0 when x=x=

We next have to make sure that this value of xx gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x)(x). A"(x)=(x)=  

Evaluate A"(x)(x) at the xx-value you gave above.

NOTE: Since your last answer is positive, this means that the graph of A(x)A(x) is concave up around that value, so the zero of A'(x)A′(x) must indicate a local minimum for A(x)A(x). 

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