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MHT CET202525 Apr 2025Morning ShiftMathematicsApplication of DerivativesActual

The rate of change of the volume of a sphere with respect to its surface area, when the radius is 5 m is

Options

  1. A5 2 ~m
  2. B2 5 ~m
  3. C1 2 ~m
  4. D1 3 ~m

Correct answer

A. 5 2 ~m

Step-by-step solution

Rate of Change of Volume with Respect to Surface Area Given a sphere of radius r and fixed formulas for volume V = 4 3 r^3 and surface area A = 4 r^2 , the derivative dV dA is determined using the chain rule as dV dA = dV/dr dA/dr . The derivatives with respect to the radius are dV dr = 4 r^2 and dA dr = 8 r . Substituting into the chain rule yields dV dA = 4 r^2 8 r = r 2 . For r = 5 m, the result is 5 2 m, which corresponds to option A .

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