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COMEDK202510 May 2025Morning ShiftMathematicsApplication of DerivativesActual

A spherical snowball is melting such that its volume is decreasing at the rate of 1 ~cm ^3 / min . The rate at which the diameter is decreasing when the diameter is 10 cm is

Options

  1. A2 75 ~cm / min
  2. B1 50 ~cm / min
  3. C11 75 ~cm / min
  4. D1 25 ~cm / min

Correct answer

B. 1 50 ~cm / min

Step-by-step solution

Let V be the volume and D be the diameter of the spherical snowball. The volume of a sphere is given by V = 4 3 r^3 , where r is the radius. Since D = 2r , we have r = D 2 . Substituting r in terms of D , V = 4 3 ( D 2 )^3 = 4 3 D^3 8 = D^3 6 . Differentiating both sides with respect to time t , we get dV dt = 6 3D^2 dD dt = D^2 2 dD dt . Given that the volume is decreasing at a rate of 1 cm ^3/ min , we have dV dt = -1 . We want to find the rate at which the diameter is decreasing, which is - dD dt when D = 10 cm

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