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COMEDK2024Evening ShiftMathematicsApplication of DerivativesActual

The rate of change of the volume of a sphere with respect to its surface area S is

Options

  1. A2 3 S
  2. BS
  3. C1 4 S
  4. D1 2 S

Correct answer

C. 1 4 S

Step-by-step solution

Let r be the radius of the sphere. The volume V and surface area S are given by V = 4 3 r^3 and S = 4 r^2 . From the expression for surface area, r^2 = S 4 , which implies r = S 4 = 1 2 S . Differentiating V with respect to r , we get dV dr = 4 r^2 . Differentiating S with respect to r , we get dS dr = 8 r . The rate of change of volume with respect to surface area is dV dS = dV/dr dS/dr = 4 r^2 8 r = r 2 . Substituting the value of r in terms of S , we get dV dS = 1 2 ( 1 2 S ) = 1 4 S . Answer: 1 4 S

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