Question

Of all closed right circular cylinders of a given volume of 100 cubic centimetres, find the dimensions of cylinder which has minimum surface area.

• A. $r={\left(\frac{50}{\pi }\right)}^{1/3}cm.$ & $h=2{\left(\frac{50}{\pi }\right)}^{1/3}cm.$
• B. $r={\left(\frac{100}{\pi }\right)}^{1/3}cm.$ & $h=2{\left(\frac{50}{\pi }\right)}^{1/3}cm.$
• C. $r={\left(\frac{50}{\pi }\right)}^{1/3}cm.$ & $h=2{\left(\frac{100}{\pi }\right)}^{1/3}cm.$
• D. None of these

Answer 1

The correct option is A $r={\left(\frac{50}{\pi }\right)}^{1/3}cm.$ & $h=2{\left(\frac{50}{\pi }\right)}^{1/3}cm.$

Volume = $\pi r^{2}h=100$
$h=\frac{100}{\pi r^2}$
Surface Area =S = $2\pi r^2+2\pi rh$
=$2\pi r(r+h)$
= $2\pi r(r+\frac{100}{\pi r^2})$
To minimise surface area
$\frac{dS}{dr}=0$
$\Rightarrow 2r-\frac{100}{\pi r^2}=0$
$\Rightarrow r=(\frac{50}{\pi }{)}^{1/3}$