AP EAMCET · Maths · Application of Derivatives
If the height of a cone of greatest volume that can be inscribed in a sphere of radius \(\mathrm{R}\) is \(\mathrm{kR}\), then ratio of the volume of the cone to the volume of the sphere is
- A \(8: 27\)
- B 27:64
- C 8:125
- D \(4: 5\)
Answer & Solution
Correct Answer
(A) \(8: 27\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \text { In } \triangle O D C \text {, } \\ & r^2=R^2-(h-R)^2 \\ & =2 h R-h^2\end{aligned}\) Hence volume of cone \(A B C=\frac{1}{3} \pi r^2 h\) \(\Rightarrow \mathrm{V}=\frac{1}{3} \pi\left(2 h R-h^2\right) h=\frac{1}{3} \pi\left(2 R h^2-h^3\right)\) Now for…
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