NCERT Class 12 · Ch 6 Exercise 6.3 · Q25
NCERT Class 12 Maths · Chapter 6 Applications Of Derivatives · Exercise 6.3 Question 25
Show that a cone with fixed slant height and maximum volume has semi-vertical angle tan⁻¹√2.
💡 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: α=tan⁻¹√2.
- With fixed slant height l, r=l sin α and h=l cos α, 0<α<π/2.
- V=(πl³/3)sin²α cos α. Its derivative has sign sin α(2cos²α−sin²α).
- The interior maximum satisfies tan²α=2; the sign changes + to − and volume vanishes at both limiting endpoints. Therefore α=tan⁻¹√2.
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