COMEDK · Maths · 27. Application of Derivatives
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 radians per second, then the radius of the base is increasing at the rate of
- A \(\dfrac{160}{3} \mathrm{~cm} / \mathrm{sec}\)
- B \(10 \mathrm{~cm} / \mathrm{sec}\)
- C \(30 \mathrm{~cm} / \mathrm{sec}\)
- D \(160 \mathrm{~cm} / \mathrm{sec}\)
Answer & Solution
Correct Answer
(A) \(\dfrac{160}{3} \mathrm{~cm} / \mathrm{sec}\)
Step-by-step Solution
Detailed explanation
\(r = h\tan\theta\) Differentiating w.r.t. time (\(h\) is constant): \(\dfrac{dr}{dt} = h\sec^2\theta \cdot \dfrac{d\theta}{dt}\) At \(\theta = 30°\): \(\sec^2 30° = \dfrac{4}{3}\), \(h = 20\) cm, \(\dfrac{d\theta}{dt} = 2\) rad/sec…
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