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KCET2024MathematicsApplication of Derivatives

The maximum volume of the right circular cone with slant height 6 units is

Options

  1. A4 3 cu units
  2. B16 3 cu units
  3. C3 3 cu units
  4. D6 3 cu units

Correct answer

B. 16 3 cu units

Step-by-step solution

Slant height of the cone, L=6 units. Let the radius be r and height be h . and volume (V)= 1 3 r^2 h= 1 3 (L^2-h^2 ) h [ L^2=r^2+h^2 ]= 1 3 (36-h^2 ) h So, d V d h = 1 3 (36-3 h^2 )=0 h=2 3 units Now, d^2 V d h^2 =-2 h 0 at h=2 3 units Hence, the maximum volume of the cone V= 1 3 (36-h^2 ) h= 1 3 (36-(2 3 )^2 ) 2 3 =16 3 cu units

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