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AP EAMCET201922 Apr 2019Morning ShiftMathematicsApplication of DerivativesActual

The maximum volume (in cubic units) of the cylinder which can be inscribed in a sphere of diameter 6 units is

Options

  1. A(12 3 )
  2. B(4 3 )
  3. C(3 3 )
  4. D(8 3 )

Correct answer

A. (12 3 )

Step-by-step solution

If (h ) is the height and (V ) be the volume, required cylinder then from the given figure. In ( O A M ), ( aligned & r^2+ ( h 2 )^2=(3)^2 & r^2+ h^2 4 =9 & r^2=9- h^2 4 & Now, volume = r^2 h= (9- h^2 4 ) h & V= (9 h- h^3 4 ) & d V d h = (9- 3 h^2 4 ) aligned ) For maximum or minimum, ( d V d h =0 ) ( array rlrl & 9- 3 h^2 4 & =0 3 h^2 4 =9 & h^2 & =12 h= 2 3 & & r^2 & =9-3 r^2=6 array ) The maximum value, ( [ d^2 V d h^2 =- 3 h 2 < 0 ] ) (= (6)(2 3 )=12 3 sq units )

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