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KVPY2010MathematicsApplication of Derivatives

Let V ₁ be the volume of a given right circular cone with O as the centre of the base and A as its apex. Let V ₂ be the maximum volume of the right circular cone inscribed in the given cone whose apex is O and whose base is parallel to the base of the given cone. Then the ratio V ₂ / V ₁ is-

Options

  1. A3 25
  2. B4 9
  3. C4 27
  4. D8 27

Correct answer

C. 4 27

Step-by-step solution

ABC and AOP are similar array c h r = H R h = rH R V ₂= 1 3 r ²( H - h )= 3 r ² H (1- r R )= H 3 ( r ³- r ³ R ) d V ₂ dr =2 r - 3 r ² R =0 r = 2 R 3 ~V _ 2 = 4 R ² H 81 array array l V ₁= R ² H 3 V ₂ ~V ₁ = 4 27 array

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