Question
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
3160 cm/sec
C
10 cm/sec
D
160 cm/sec
Answer
3160 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
⇒ dtdr = 20 sec2 θ dtdθ
⇒dtdr = 20 sec2 30° × 2 [∵θ=30∘anddtdθ=2]
⇒ dtdr = 20 × 34 × 2 cm/sec = 3160 cm/sec
Hence (2) is the correct answer.