The altitude of a cone is 20 cm and its semi-vertical angle is `30^0` . If the semi-vertical angle is increasing at the rate of `2^0`  per second, then the radius of the base is increasing at the rate of (a) 30 cm/sec     (b) `160/3` cm/sec      (c) 10 cm/sec    (d) 160 cm/sec

Options

A. 30 cm/sec
B. `(160)/(3)cm//sec`
C. 10 cm/sec
D. 160 cm/sec

Answer

B

Explanation

Let `theta` be the semi-vertical angle and r be the radius of the cone at time t. Then,

`r=20 tan theta`
`implies (dr)/(dt)=20 sec^(2)theta(d theta)/(dt)`
`implies (dr)/(dt)=20 sec^(2) 30^(@)xx2" "[because theta=30^(@) and (d theta)/(dt)=2"(given)"]`
`implies (dr)/(dt)=20xx(4)/(3)xx2cm//sec=(160)/(3)cm//sec`