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