Question

Score! 07120 U12 drswereu Question 1 < > 10 pts 5 1 28 Details Textbook Videos [+] A box with a square base and open top must have a volume of 364500 cm. 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 2, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of z.] Simplify your formula as much as possible. A2) = Next, find the derivative, A'(2). A'(x) = Now, calculate when the derivative equals zero, that is, when A'(x) = 0. (Hint: multiply both sides by ? .] A'(x) = 0 when I = We next have to make sure that this value of gives a minimum value for the surface area. Let's use the second derivative test. Find A (2). A"(2) = Evaluate A"(2) at the C-value you gave above. NOTE: Since your last answer is positive, this means that the graph of A(z) is concave up around that value, so the zero of A'() must indicate a local minimum for A2). (Your boss is happy now.) Question Help: Message instructor Submit Question

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Answer 1

Let at be the length of one side of the aquare and 'n be the beight of the box. x ²b Volume > 364500 = x²h b. 364500 ge² A Coc) = 1² - 4ach 1² + 4xx 364500 Surface area I? 2 90'+ 1458000 16800 A C) = 2x+ - 1458000xł -2 = 200- 1458000 Ic2 A'(x) = 0 5 232-1458000 अ२ 2x 1458000 12

-> 2003 = 1458000 3 N = 729000 -> X = 90 -3 A" (36) = 2 2 - 1458000X-20 = 2 + 2916000 3 00 A" (90) - 2+ 2916000 3 1) is concave up 2+4 6, which is positive means that the graph of AG) around that value, so of A'Cal is a local minimum for A(x). This