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COMEDK202510 May 2025Morning ShiftMathematicsApplication of DerivativesActual

Oil from a conical funnel is dripping at the rate of 5 ~cm ^3 / s . If the radius and height of the funnel are 10 cm and 20 cm respectively, then the rate at which the oil level drops when it is 5 cm from the top is

Options

  1. A- 4 45 ~cm / s
  2. B- 2 45 ~cm / s
  3. C- 4 45 ~cm / s
  4. D8 45 ~cm / s

Correct answer

A. - 4 45 ~cm / s

Step-by-step solution

Let R = 10 cm be the radius and H = 20 cm be the height of the conical funnel. Let r and h be the radius and height of the oil level at any time t . By similar triangles, r h = R H = 10 20 = 1 2 , so r = h 2 . The volume of the oil in the funnel is V = 1 3 r^2 h = 1 3 ( h 2 )^2 h = h^3 12 . Differentiating with respect to time t , we get dV dt = 12 3h^2 dh dt = h^2 4 dh dt . Given dV dt = -5 cm ^3 /s (since the volume is decreasing). The oil level is 5 cm from the top, so the height of the oil from the vertex is h

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