An open-top box is to be constructed from a 8 in by 10 in rectangular sheet of t

Question

An open-top box is to be constructed from a 8 in by 10 in rectangular sheet of tin by cutting out squares of equal size at each corner, then folding up the resulting flaps. Let x denote the length of the side of each cut-out square. Assume negligible thickness. (a) Find a formula for the volume, V, of the box as a function of x. V(x)= (b) For what values of x does the formula from part (a) make sense in the context of the problem? help (inequalities) (c) On a separate piece of paper, sketch a graph of the volume function. (d) What, approximately, is the maximum volume of the box?

Explanation / Answer

Volume = x(6-2x)(8-2x) Volume = (6x-2x^2)(8-2x) Volume = 48x - 12x^2 -16x^2 + 4x^3 = 4x^3 - 28x^2 +48x d Volume / dx = 12x^2 - 56x + 48 = 0 6x^2 - 28x + 24 = 0 3x^2 - 14x + 12 = 0 x = 3.5352 or 1.1315 Second Derivative Test: d²V / dx² = 24x - 56 d²V / dx²(x = 3.5352) = 24(3.5352) - 56 = 28.8448 > 0, thus a min. (not what we are looking for.) d²V / dx²(x = 1.1315) = 24(1.1315) - 56 = -28.844 < 0, thus a max.(what we are looking for.) Going back to volume: Volume = x(6-2x)(8-2x) Volume = (1.1315)(6 - 2(1.13151))(8 - 2(1.1315))=24.26

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