A bucket weighs $5 \mathrm { lb }$ and a rope of negligible weight are used to draw water from a well that is 40 ft deep. The bucket starts with $50 \mathrm { lb }$ of water and is pulled up at a rate of $10 \mathrm { ft } / \mathrm { s }$, but water leaks out of a hole in the bucket at a rate of $0.5 \mathrm { lb } / \mathrm { s }$. Find the work done in pulling the bucket to the top of the well.
A) $2,160 \mathrm { ft } - \mathrm { lb }$
B) $2,660 \mathrm { ft } - \mathrm { lb }$
C) $3,660 \mathrm { ft } - \mathrm { lb }$
D) $2,260 \mathrm { ft } - \mathrm { lb }$
E) $3,160 \mathrm { ft } - \mathrm { lb }$

Question

A bucket weighs $5 \mathrm { lb }$ and a rope of negligible weight are used to draw water from a well that is 40 ft deep. The bucket starts with $50 \mathrm { lb }$ of water and is pulled up at a rate of $10 \mathrm { ft } / \mathrm { s }$, but water leaks out of a hole in the bucket at a rate of $0.5 \mathrm { lb } / \mathrm { s }$. Find the work done in pulling the bucket to the top of the well. 
A) $2,160 \mathrm { ft } - \mathrm { lb }$
B) $2,660 \mathrm { ft } - \mathrm { lb }$
C) $3,660 \mathrm { ft } - \mathrm { lb }$
D) $2,260 \mathrm { ft } - \mathrm { lb }$
E) $3,160 \mathrm { ft } - \mathrm { lb }$

Quizplus · Accepted Answer

The work done is equal to the force required to pull the bucket up the well multiplied by the distance it is pulled. The force required to pull the bucket up is equal to the weight of the bucket and the water in it minus the force due to the water leaking out. The weight of the bucket and the water in it is $50 \mathrm { lb } + 5 \mathrm { lb } = 55 \mathrm { lb }$. The force due to the water leaking out is $0.5 \mathrm { lb } / \mathrm { s } \times 10 \mathrm { ft } / \mathrm { s } = 5 \mathrm { lb }$. Therefore, the force required to pull the bucket up is $55 \mathrm { lb } - 5 \mathrm { lb } = 50 \mathrm { lb }$. The distance the bucket is pulled is $40 \mathrm { ft }$. Therefore, the work done is $50 \mathrm { lb } \times 40 \mathrm { ft } = 2,000 \mathrm { ft } - \mathrm { lb }$. However, we also need to account for the work done against gravity as the water leaks out. The weight of the water that leaks out is $0.5 \mathrm { lb } / \mathrm { s } \times 10 \mathrm { s } = 5 \mathrm { lb }$. The distance the water leaks out is $40 \mathrm { ft }$. Therefore, the work done against gravity is $5 \mathrm { lb } \times 40 \mathrm { ft } = 200 \mathrm { ft } - \mathrm { lb }$. Therefore, the total work done is $2,000 \mathrm { ft } - \mathrm { lb } + 200 \mathrm { ft } - \mathrm { lb } = 2,200 \mathrm { ft } - \mathrm { lb }$.

A Bucket Weighs 5lb5 \mathrm { lb }5lb And a Rope of Negligible Weight Are Used to Draw

A Bucket Weighs $5 \mathrm { lb }$ And a Rope of Negligible Weight Are Used to Draw