Question

The altitude of a cone is $20cm$ and its semi-vertical angle is ${30}^0$. If the semi-vertical angle is increasing at the rate of $2^0$ per second, then the radius of the base is increasing at the rate of

• A. $30cm/sec$
• B. $\frac{160}{3}cm/sec$
• C. $10cm/sec$
• D. $160cm/sec$

Answer 1

The correct option is B $\frac{160}{3}cm/sec$

Here,$\frac{r}{h}=\tan \alpha$
$\frac{r}{20}=\tan {30}^0$
$\Rightarrow r=\frac{20}{\sqrt{3}}cm$
Since, $r=20\tan \alpha$
$\frac{dr}{dt}=20se{c}^2\alpha \frac{d\alpha }{dt}$
$\Rightarrow \frac{dr}{dt}=\frac{160}{3}cm/sec$