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KVPY2017MathematicsApplication of Derivatives

A solid hemisphere is mounted on a solid cylinder, both having equal radii. If the whole solid is to have a fixed surface area and the maximum possible volume, then the ratio of the height of the cylinder to the common radius is

Options

  1. A1: 1
  2. B1: 2
  3. C2: 1
  4. D2 : 1

Correct answer

A. 1: 1

Step-by-step solution

array l S =2 R ²+2 Rh + R ² ( R = radius of hemisphere & cylinder) V = 2 3 R ³+ R ² ~h ~V = 2 3 R ³+ R ² ( 5-3 R ² 2 R ) dV dR =2 R ²+ 5 2 - 9 2 R ² array for maximum & minimum dV dR =0 array l 5 R²=S 5 R²=3 R²+2 R h 2 R=2 h h: R=1: 1 array

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