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An open box with a square base to be made out of a given cardb

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An open box with a square base to be made out of a given cardboard of area $c^{2}$ (square) units. Show that the maximum volume of the box is $\frac{c ^{3}}{6 3}$ (cubic units.)

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An open box with a square base to be made out of a given cardboard of area

c^{2}

(square) units. Show that the maximum volume of the box is

\frac{c ^{3}}{6 3}

(cubic units.)

11 mins ago

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Text solutionVerified

Let each side of the base be

a

and height be

h

. Then,

c^{2} = (a^{2} + 4 ah) \Rightarrow h = \frac{( c ^{2} - a ^{2} )}{4 a}

So,

V = a^{2} \times h \Rightarrow V = \frac{( c ^{2} a - a ^{3} )}{4}

∴ \frac{d V}{d a} = \frac{( c ^{2} - 3 a ^{2} )}{4}

and

\frac{d ^{2} V}{d a ^{2}} = - \frac{3}{2} a < 0

Thus,

V

is maximum when

a^{2} = \frac{c ^{2}}{3}

and

h = \frac{c}{2 3}

∴ V (ma x) = a^{2} \times h = \frac{c ^{3}}{6 3}

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Question Text	An open box with a square base to be made out of a given cardboard of area $c^{2}$ (square) units. Show that the maximum volume of the box is $\frac{c ^{3}}{6 3}$ (cubic units.)
Answer Type	Text solution:1
Upvotes	150