Question

An open rectangular tank with a square base and $32c.c$. of capacity has least surface area in $sq.cms$. is

• A. $48$
• B. $16$
• C. $32$
• D. $12$

Answer 1

The correct option is A $48$

Volume of tank $V=x^{2}h$
$\Rightarrow 32=x^{2}h$
Surface area of open tank $S=x^2+4xh$
$\Rightarrow S=x^2+\frac{128}{x}$
$\Rightarrow S^{\prime }\left(x\right)=2x-\frac{128}{x^2}$
For maxima or minima,
$S^{\prime }\left(x\right)=0$
$\Rightarrow x=4$
$S^{\prime \prime }\left(x\right)=2+\frac{256}{x^3}>0$ at $x=4$
Hence, least surface area $=16+32=48$ sq.units