A box with a square base and open top must have a volume of48668 cm3. We wish to
ID: 1889220 • Letter: A
Question
A box with a square base and open top must have a volume of48668 cm3. We wish to find the dimensions of the box that 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) Now, calculate when the derivative equals zero, that is: when A'(X) = 0.[Hint multiply both sides by x2.] 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 x -value you gave above. NOTE: Since your last answer is positive, this means that the graph of A(x) is concave up around that value, so the zero of A'(X) must indicate a local minimum for A(x). (Your boss is happy now.)Explanation / Answer
x^2 * h = 48668 x^2 + 4*xh = A(x) A(x) = x^2 + 4*48668 / x A'(x) = 2x - 4*48668 / x^2 => x = 46 A''(x) = 2 + 2*4*48668 / x^2 A''(46) > 0 So minimum
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