NCERT Class 12 · Ch 6 Exercise 6.3 · Q24
NCERT Class 12 Maths · Chapter 6 Applications Of Derivatives · Exercise 6.3 Question 24
Show that the cone of given volume and least curved surface has height √2 times base radius.
💡 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=√2 r.
- Let K=3V/π>0, so h=K/r². Curved area L=πr√(r²+h²). Minimize its square L²=π²(r⁴+K²/r²), r>0.
- The derivative is π²(4r³−2K²/r³), which vanishes at K²=2r⁶.
- Its sign changes from negative to positive there, and L² tends to infinity at both domain ends. Thus h²=K²/r⁴=2r² and h=√2r.
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