Explanation Given: The mass of the bucket is 10 kg. The height of lift is 12 m. The mass of the rope is 0.8 kg/m . The weight of water in the bucket at initial stage is 36 kg. The leaked bucket gets empty at the height of 12 m. The draining of the water from the bucket occurs at a constant rate. Write the expression to find the work done W to pump the water from a to b , when a force f ( x ) is applied on it. W = lim n → ∞ ∑ i = 1 n f ( x i ∗ ) Δ x = ∫ a b f ( x ) d x (1) Calculation: Consider the height of the bucket at any instant of time is x m from the ground. ( 0 < x < 12 ) Calculate the mass of the rope using the relation: Mass of the rope = Unit mass of the rope × length of rope = 0.8 kg m × ( 12 − x ) m = 0.8 ( 12 − x ) kg The water in the bucket drains at a constant rate. The mass of water in bucket is 36 kg when the bucket is at the ground level. The mass of water in bucket is 0 kg when the bucket is roped at 12 m from the ground. Consider the mass of the water in the bucket per metre lift of bucket is m w . m w = 36 12 kg m Calculate the mass of the water in the bucket, when the bucket is at a height of 12 m from ground level. Mass of the water = Mass of water per metre lift of bucket × height of lift of bucket = 36 12 kg m × ( 12 − x ) m = 36 12 ( 12 − x ) kg Calculate the total mass of rope and the bucket with water at a height of x from the ground is given by the relation. Total mass = Mass of the water+Mass of the bucket+Mass of water = 36 12 ( 12 − x ) kg + 10 kg+ 0.8 ( 12 − x ) kg = 3 .8 ( 12 − x ) + 10 kg = 55

Single Variable Calculus: Early Transcendentals

8th Edition

ISBN: 9781305270336

Author: James Stewart

Publisher: Cengage Learning

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Chapter 6.4, Problem 17E

To determine

To express: The work done to lift the leaky bucket with water to a height of 12 m as Riemann sum.

To calculate: The work done to lift the leaky bucket with water to a height of 12 m.

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Chapter 6.4, Problem 16E

Chapter 6.4, Problem 18E

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Single Variable Calculus: Early Transcendentals

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To express: The work done to lift the leaky bucket with water to a height of 12 m as Riemann sum. To calculate: The work done to lift the leaky bucket with water to a height of 12 m.