NCERT Class 12 · Ch 6 Miscellaneous Exercise · Q14
NCERT Class 12 Maths · Chapter 6 Applications Of Derivatives · Miscellaneous Exercise Question 14
Find the height and volume of the maximum-volume cylinder inscribed in a sphere of radius R.
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: h=2R/√3; Vmax=4πR³/(3√3).
- The axial section gives r²+(h/2)²=R², so V=πh(R²−h²/4), 0<h<2R.
- V′=π(R²−3h²/4), zero at h=2R/√3. It is positive before and negative after; endpoint volumes vanish.
- Then r²=2R²/3 and Vmax=π(2R²/3)(2R/√3)=4πR³/(3√3).
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