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

Answer 1

Answer: (B)

Let $\theta$ be the semi-vertical angle and $r$ be the radius of the cone at time $t$.
Then, $r=20\tan \theta$
$\Rightarrow \frac{dr}{dt}=20{\sec }^2\theta \frac{d\theta }{dt}$
$\Rightarrow \frac{dr}{dt}=20{\sec }^{2}30^{\circ }\times 2[\because \theta =30^{\circ }$ and $\frac{d\theta }{dt}=2]$
$\Rightarrow \frac{dr}{dt}=20\times \frac{4}{3}\times 2cm/s=\frac{160}{3}cm/s$