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Grade 12Differential Calculus

An open box with a square base is to be made out of a given quantity of sheet of area a^2.
Show that: The max volume of the box is a^3/6*(3^1/2)

Faiz

12 Years agoGrade 12

1 Answer

Jitender Singh

Approved Tutor Answer12 Years ago

Ans:

Let side of the square box is p & height is q

Volume V:

$p^{2}q$

Area of box:

$p^{2}+pq+pq+pq+pq = p^{2}+4pq=a^{2}$

$q=\frac{a^{2}-p^{2}}{4p}$

$V=\frac{a^{2}-p^{2}}{4p}.p^{2}$

$V=\frac{a^{2}-p^{2}}{4}.p$

$\frac{\partial V}{\partial p} = 0$

$a^{2}-3p^{2}=0$

$p^{2}=\frac{a^{2}}{3}$

$q=\frac{a^{2}-\frac{a^{2}}{3}}{4\frac{a}{\sqrt{3}}} = \frac{a}{2\sqrt{3}}$

Max. Volume:

$\frac{a^{2}}{3}.\frac{a}{2\sqrt{3}} = \frac{a^{3}}{6\sqrt{3}}$

Thanks & Regards

Jitender Singh

IIT Delhi

askIITians Faculty

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