Q.32 A box with a square base is to have an open top. The surface area of box is 147 sq.cm. What should be its dimensions in order that the volume is largest?
Best answer
Answer: Base 7 cm × 7 cm and height 3.5 cm (largest volume 171.5 cu. cm).
Let the side of the square base be \(x\) cm and the height \(h\) cm. With an open top, the surface area is \(x^2 + 4xh = 147\), so \(h = \frac{147 - x^2}{4x}\).
Volume \(V = x^2 h = x^2 \cdot \frac{147 - x^2}{4x} = \frac{147x - x^3}{4}\).
\(\frac{dV}{dx} = \frac{147 - 3x^2}{4}\) and \(\frac{d^2V}{dx^2} = -\frac{6x}{4} = -\frac{3x}{2}\).
For extreme values, \(\frac{dV}{dx} = 0 \Rightarrow 3x^2 = 147 \Rightarrow x^2 = 49 \Rightarrow x = 7\) (since \(x \gt 0\)).
At \(x = 7\), \(\frac{d^2V}{dx^2} = -\frac{21}{2} \lt 0\), so \(V\) is maximum.
\(h = \frac{147 - 49}{28} = \frac{98}{28} = 3.5\) cm.
So the box should have a square base of side 7 cm and height 3.5 cm. Maximum volume \(= 7^2 \times 3.5 = 171.5\) cu. cm.
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