A box a square base and square top is to be made from 400 $m^{2}$ of number. If its volume is maximum, then the ratio of an edge of the base and height is equal to

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Solution

Let edge be $a$ and height be $h$
Surface area $= 2 (a^{2} + a h + a h) = 400 a^{2} + 2 a h = 200 h = \frac{200 - a^{2}}{2 a} V = a^{2} \times h = a^{2} \times \frac{(200 - a^{2})}{2 a} V = \frac{200 a - a^{3}}{2} \frac{d V}{d a} = \frac{1}{2} [200 - 3 a^{2}] \frac{d V}{d a} = 0, a = \sqrt{\frac{200}{3}}, h = \frac{10 \sqrt{6}}{3} = \frac{a}{h} = 1$

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A box a square base and square top is to be made from 400 m2 of number. If its volume is maximum, then the ratio of an edge of the base and height is equal to

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A box a square base and square top is to be made from 400 $m^{2}$ of number. If its volume is maximum, then the ratio of an edge of the base and height is equal to