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Question Number 61322 by Tawa1 last updated on 31/May/19

Answered by MJS last updated on 02/Jun/19

the cuboid with the greatest volume at a  given surface is a cube    Surface=2a^2 +4ab=216 ⇒ b=((54)/a)−(a/2)  Volume=a^2 b=54a−(1/2)a^3   V′=54−(3/2)a^2 =0 ⇒ a=6 ⇒ b=6

$$\mathrm{the}\:\mathrm{cuboid}\:\mathrm{with}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{volume}\:\mathrm{at}\:\mathrm{a} \\ $$$$\mathrm{given}\:\mathrm{surface}\:\mathrm{is}\:\mathrm{a}\:\mathrm{cube} \\ $$$$ \\ $$$${S}\mathrm{urface}=\mathrm{2}{a}^{\mathrm{2}} +\mathrm{4}{ab}=\mathrm{216}\:\Rightarrow\:{b}=\frac{\mathrm{54}}{{a}}−\frac{{a}}{\mathrm{2}} \\ $$$${V}\mathrm{olume}={a}^{\mathrm{2}} {b}=\mathrm{54}{a}−\frac{\mathrm{1}}{\mathrm{2}}{a}^{\mathrm{3}} \\ $$$${V}'=\mathrm{54}−\frac{\mathrm{3}}{\mathrm{2}}{a}^{\mathrm{2}} =\mathrm{0}\:\Rightarrow\:{a}=\mathrm{6}\:\Rightarrow\:{b}=\mathrm{6} \\ $$

Commented by Tawa1 last updated on 02/Jun/19

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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