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Optimization problems with an open-top box — Krista King Math | Online math help

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A rectangular box, open at the top, is to have a volume of 32 c... | Filo

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c) |\begin{tabular} { l } The rectangular box, open at top, is... …

A rectangular box, open at the top, is to have a volume of 32 cubic feet. What must be the dimensions so that the total surface is a minimum? If x, y and z are the edges (see Fig. 8-4), then (1) Volume of box =V=x y z=32 (2) Surface area of box =S=x y 2 y z 2 x z or, since z=32 / x y from (1), S=x y (64)/(x) (64)/(y) (∂S)/(∂x)=y-(64)/(x^2)=0 when (s) x^2 y=64, (∂S)/(∂y)=x-(64)/(y^2)=0 when (4) x y^2=64 Dividing equations ( 3) and (4), we find y=x so that x^3=64 or x=y=4 and z=2. For x=y=4, Δ

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