TS EAMCET · Maths · Application of Derivatives
A right circular cone is inscribed in a sphere of radius 3 units. If the volume of the cone is maximum, then semi vertical angle of the cone is
- A \(\frac{\pi}{4}\)
- B \(\frac{\pi}{6}\)
- C \(\tan ^{-1}(\sqrt{2})\)
- D \(\tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)\)
Answer & Solution
Correct Answer
(D) \(\tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & V=\frac{1}{3} \pi r^2 h=\frac{1}{3} \pi \times\left(9-x^2\right)(3+x) \\ & \Rightarrow V=\frac{\pi}{3}\left(27+9 x-3 x^2-x^3\right) \\ & \frac{d V}{d x}=\frac{\pi}{3}\left(9-6 x-3 x^2\right)=0 \Rightarrow x^2+2 x-3=0 \\ & x=1 \quad(x \neq-3) \\ & \frac{d^2 V}{d…
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