Question

A bell tent consists of a conical portion above a cylindrical portion near the ground. For a given volume and a circular base of a given radius, the amount of the canvas used is minimum when the semi-vertical angle of the cone is

• A. ${\cos }^{-1}\frac{2}{3}$
• B. ${\sin }^{-1}\frac{2}{3}$
• C. ${\cos }^{-1}\frac{1}{3}$
• D. None of these

Answer 1

The correct option is A ${\cos }^{-1}\frac{2}{3}$


$V=$ Volume of cone $+$ Volume of cylinder
$=\frac{\pi }{3}{r}^{2}h+\pi r^{2}H$
$V=\frac{\pi }{3}{r}^2(h+3H)$
$\Rightarrow H=\frac{\frac{3V}{\pi r^2}-h}{3}$
Now, surface area, $S=\pi rl+2\pi rH=\pi r\sqrt{h^2+r^2}+2\pi r\times \left(\frac{\frac{3V}{\pi r^2}-h}{3}\right)$
Now,$\frac{dS}{dh}=0$
$\Rightarrow \pi r\frac{h}{\sqrt{h^2+r^2}}-\frac{2\pi r}{3}=0$
$\Rightarrow \frac{h}{\sqrt{h^2+r^2}}=\frac{2}{3}$
$\Rightarrow 5h^2=4r^2$
$\Rightarrow \frac{r}{h}=\frac{\sqrt{5}}{2}=\tan \theta$
$\Rightarrow \cos \theta =\frac{2}{3}\Rightarrow \theta ={\cos }^{-1}\frac{2}{3}$