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Class 12 Applications Of Derivatives · Long Answer3 Marksgeometric_optimization

Class 12 Maths · Applications Of Derivatives · Long Answer #98

Prove that a right circular cone of fixed positive volume and least curved surface area has semi-vertical angle arccot(√2), using arccot in (0,π).
Right circular cone: axial section
Right circular cone: axial section

✅ Step-by-Step Solution

Final Answer: θ=arccot(√2)
  1. For fixed volume V, h=3V/(πr²) and the square of curved area is F=π²r⁴+9V²/r².
  2. F′=4π²r³−18V²/r³=0 yields 4π²r⁶=18V²; F″=12π²r²+54V²/r⁴>0 proves the global minimum.
  3. Substitute V=πr²h/3 into that equation: h²=2r², so h/r=√2.
  4. For the semi-vertical angle θ∈(0,π/2), cotθ=h/r=√2. Therefore θ=arccot(√2).

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] For fixed volume V, h=3V/(πr²) and the square of curved area is F=π²r⁴+9V²/r².
  • [1 mark] F′=4π²r³−18V²/r³=0 yields 4π²r⁶=18V²; F″=12π²r²+54V²/r⁴>0 proves the global minimum.
  • [1 mark] Substitute V=πr²h/3 into that equation: h²=2r², so h/r=√2.
  • [0.5 marks] For the semi-vertical angle θ∈(0,π/2), cotθ=h/r=√2. Therefore θ=arccot(√2).

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