Question

The altitude of a cone is $20\, cm$ 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 and $r$ be the radius of the cone at time $t$.
Then, $r=20 \tan \theta$
$\Rightarrow \frac{dr}{dt}=20 \sec ^{2} \theta \frac{d \theta}{dt}$
$\Rightarrow \frac{d r}{d t}=20 \sec ^{2} 30^{\circ} \times 2\left[\because \theta=30^{\circ}\right.$ and $\left.\frac{d \theta}{d t}=2\right]$
$\Rightarrow \frac{d r}{d t}=20 \times \frac{4}{3} \times 2 \,cm / s =\frac{160}{3}\, cm / s$