Question

A bucket that weighs 5 lb and a rope of negligible weight are used to draw water from a well that is 70 ft deep. The bucket is filled with 38 lb of water and is pulled up at a rate of 2.5 ft/s, but water leaks out of a hole in the bucket at a rate of 0.25 lb/s. Find the work done in pulling the bucket to the top of the well.

Show how to approximate the required work by a Riemann sum. (Let x be the height in feet above the bottom of the well. Enter xᵢ* as xᵢ.)

n
lim ∑ ( ____ ) Δx
n ->[infinity] i = 1

Answer 1

The Physics question asks for the calculation of the work done in pulling up a leaking bucket from a well. It requires integrating the changing weight of the bucket over the distance lifted, considering the rate of water leaking out.

Step-by-step explanation:

The subject of the student's question is Physics, specifically, it deals with the concept of work done in mechanics. When calculating the work done in pulling the bucket to the top of the well, the varying weight of the bucket due to the leaking water must be taken into account. To find the work done, we can integrate the weight of the bucket with respect to distance. Since the bucket's weight changes as water leaks, the work done is the integral of the force exerted on the bucket, which varies linearly with time, over the distance it is lifted.

Ignoring the bucket's variable weight due to water leakage would result in calculating the work done as simply force times distance, which would not be correct in this situation. Due to the leakage, the force required to lift the bucket decreases over time, making the calculation slightly more complex. It requires setting up an equation that accounts for the change in weight as a function of time and integrates this over the distance lifted.