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Question: The altitude of a cone is 20 cm and its semi-vertical angle is 30°. If the semi-vertical angle is in...

The altitude of a cone is 20 cm and its semi-vertical angle is 30°. If the semi-vertical angle is increasing at the rate of 2° per second, then the radius of the base is increasing at the rate of –

A

30 cm/sec

B

1603\frac{160}{3} cm/sec

C

10 cm/sec

D

160 cm/sec

Answer

1603\frac{160}{3} cm/sec

Explanation

Solution

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

r = 20 tan θ

drdt\frac{dr}{dt} = 20 sec2 θ dθdt\frac{d\theta}{dt}

drdt\frac{dr}{dt} = 20 sec2 30° × 2 [θ=30anddθdt=2]\left\lbrack \because\theta = 30{^\circ}and\frac{d\theta}{dt} = 2 \right\rbrack

⇒  drdt\frac{dr}{dt} = 20 × 43\frac{4}{3} × 2 cm/sec = 1603\frac{160}{3} cm/sec

Hence (2) is the correct answer.