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

The correct option is B

$\frac{160}{3}cm/sec$

Let $\theta$ be the semi-vertical angle and r be the radius of the cone at time t. Then,
$r=20tan\theta$
$\Rightarrow \frac{dr}{dt}=20se{c}^2\theta \frac{d\theta }{dt}$

$\Rightarrow \frac{dr}{dt}=20se{c}^2{30}^{\circ }\times 2[\because \theta ={30}^{\circ }and\frac{d\theta }{dt}=2]\Rightarrow \frac{dr}{dt}=20\times \frac{4}{3}\times 2cm/sec$
Hence (b) is the correct answer