NCERT Class 12 · Ch 6 Exercise 6.3 · Q23
NCERT Class 12 Maths · Chapter 6 Applications Of Derivatives · Exercise 6.3 Question 23
Prove the largest cone inscribed in a sphere of radius R has volume 8/27 of the sphere volume.
💡 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.
Master Chapter 6: Applications Of Derivatives
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Applications Of Derivatives.
← Previous Question
Cut 28 m wire into a square perimeter and circle circumference to…
Next Question →Show that the cone of given volume and least curved surface has…
All Questions in Applications Of Derivatives — Exercise 6.3
Related Questions in Applications Of Derivatives
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximal open interval on which f(x)=x²e^(−x) has positive derivative.
Class 12 Applications Of Derivatives · Mcq Single Correct
For f(x)=x²−4x+6, identify the maximal open interval where f′(x)>0.
Class 12 Applications Of Derivatives · Mcq Single Correct
For x>0, at which x is f(x)=x^x stationary?
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximum value of (1/x)^x for x>0.
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximum of [x(x−1)+1]^(1/3) on 0≤x≤1.
Class 12 Applications Of Derivatives · Mcq Single Correct
A trapezium area model is A(x)=(10+x)√(100−x²) cm² for 0≤x≤10. Find its maximum.