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AP EAMCET2010MathematicsApplication of Derivatives

The height of the cone of maximum volume inscribed in a sphere of radius R is

Options

  1. AR 3
  2. B2 R 3
  3. C4 R 3
  4. D4 R 3

Correct answer

C. 4 R 3

Step-by-step solution

Let the height of the cone =h and the radius of the cone =r Given, radius of the sphere =R Now, In O P B R^2=r^2+(h-R)^2 r^2=R^2-(h-R)^2=(R+h-R)(R-h+R) r^2=h(2 R-h) The volume of the cone is V= 1 3 r^2 h V= 1 3 h(2 R-h) h V= 3 (2 R h^2-h^3 ) Differentiating with r to h d V d h = 3 (4 R h-3 h^2 ) For maximum or minimum value of volume d V d h =0 3 (4 R h-3 h^2 )=0 h(4 R-3 h)=0 h=0, h= 4 R 3 (Not possible) Now, d^2 V d h^2 = 3 (4 R-6 h) ( d^2 V d h^2 )_ ( at h= 4 R 3 ) = 3 (4 R-6 4 R 3 )= 3 (4 R-8 R)=- 4 3 R Negative

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