Question

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

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

Answer 1

Answer: (B)

Let $\theta$ be the semi-vertical angle angle and $r$ be the radius of the cone at time $t$.
Then, $r=20\tan \theta$
$\frac{dr}{dt}=20{\sec }^2\theta \frac{d\theta }{dt}$
$\Rightarrow \frac{dr}{dt}=20\times {\sec }^{2}30^{\circ }\times 2$
$=\frac{160}{3}cm/sec$