A rectangular box (with top) is to have volume 800 in 3, and it base is to have

Question

A rectangular box (with top) is to have volume 800 in 3, and it base is to have length and such that the length is 4 time the width. What is the minimum possible surface area of such a box? Show your work Use calculus and than verify with the calculator. I am expecting to see the Volume formula, the Surface area formula, the derivative of the appropriate formula and the ordered pair for the solution. As well as the work associated with these formula. I also expect to see neat and readable work. a final answer with correct labels, and a graph of the appropriate function.

Explanation / Answer

let l be the length, b be the width & h be the height l = 4b volume = lbh = 800 4b^2h = 800 b^2h = 200 h = 200/b^2 surface area, A = 2(lb+bh+hl) A = 2(4b^2 + b.200/b^2 + 200/b^2.4b) A = 8b^2 + 400/b + 1600/b A = 8b^2 + 2000/b dA/db = 16b - 2000/b^2 = 0 16b = 2000/b^2 b^3 = 2000/16 b^3 = 1000/8 b = 10/8 = 5/4 l = 4b = 5 h = 200/b^2 = 200/25/4 = 32 hence A = 8b^2 + 2000/b A = 8.25/16 + 2000/5/4 A = 25/2 + 1600 A = 3225/2

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