NCERT Class 12 · Ch 6 Miscellaneous Exercise · Q12
NCERT Class 12 Maths · Chapter 6 Applications Of Derivatives · Miscellaneous Exercise Question 12
Show the maximum-volume cone inscribed in a sphere of radius R has altitude 4R/3.
💡 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=4R/3; Vmax=32πR³/81=(8/27)(4πR³/3).
- Take the apex at the sphere top and cone height h, 0<h<2R. The base radius satisfies r²=R²−(R−h)²=2Rh−h².
- Thus V(h)=(π/3)h²(2R−h). Its derivative is (π/3)h(4R−3h).
- The derivative is positive before h=4R/3 and negative after it, and both limiting endpoint volumes are zero.
- Substitute h=4R/3: r²=8R²/9 and Vmax=32πR³/81. Dividing by 4πR³/3 gives 8/27.
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