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BITSAT2018MathematicsApplication of DerivativesActual

An inverted conical flask is being filled with water at the rate of 3 cm 3 sec - 1 . The height of the flask is 10 cm and the radius of the base is 5 cm . How fast is the water level rising when the level is 4 cm ?

Options

  1. A4 3 π   cm   sec - 1
  2. B3 4 π   cm   sec - 1
  3. C3 π 4   cm   sec - 1
  4. D4 3 π   cm   sec - 1

Correct answer

B. 3 4 π   cm   sec - 1

Step-by-step solution

Let depth of water at time t be h and the radius of the base of water level be r . From similar ∆ O M P and ∆ O N Q , we have O M O N = P M Q N ⇒ h 10 = r 5 ⇒ r = 5 h 10 ⇒ r = h 2 We have, V = 1 3 π r 2 h = 1 3 π h 2 2 h ⇒ V = π h 3 12 On differentiating both sides, we get d V d t = π 12 · 3 h 2 d h d t = π h 2 4 d h d t Given, d V d t = 3   cm 3   sec - 1 when h = 4   cm , so we get 3 = π × 4 2 4 d h d t ⇒ d h d t = 3

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