A bucket that weighs 4 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 44 lb of water and is pulled up at a rate of 2 ft/s, but water leaks out of a hole in the bucket at a rate of 0.2 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.
well, the weight is 44-0.2t
now integrate that over the required distance.
To find the work done in pulling the bucket to the top of the well, we need to calculate the total work done against gravity while pulling the bucket.
The work done against gravity is given by the formula:
Work = Force * Distance
Here, the force is the weight of the bucket (4 lb) plus the weight of the water in the bucket (44 lb), and the distance is the height of the well (70 ft).
Work = (Weight of bucket + Weight of water) * Height of well
Work = (4 lb + 44 lb) * 70 ft
Work = 48 lb * 70 ft
Now, let's calculate the approximate work by a Riemann sum. We'll break down the distance of 70 ft into small intervals and calculate the work done for each interval.
Let's choose a small interval of length Δx and divide the height of the well into n equal intervals.
Δx = 70 ft / n
In each interval, the weight of the bucket and water will be different due to water leakage.
Let's assume at each interval, the weight of the water in the bucket remains constant.
Weight of water in each interval = 44 lb
The weight of the bucket will be reduced due to water leakage at a rate of 0.2 lb/s. The weight of the bucket at each interval can be calculated as follows:
Weight of bucket = Initial weight - Leakage rate * time
Here, the initial weight of the bucket is 4 lb, and the time can be approximated by the midpoint of each interval.
The work done in each interval is given by:
Work interval = (Weight of bucket + Weight of water) * Δx
To calculate the total work, we need to calculate the work done in each interval and sum them up using a Riemann sum.
Total work ≈ Σ (Work interval)
Approximating the required work by a Riemann sum will give us an estimate of the work done in pulling the bucket to the top of the well.
To find the work done in pulling the bucket to the top of the well, we need to calculate the total amount of work done against gravity.
The work done is given by the equation: Work = Force * Distance
In this case, the force is the weight of the bucket and water, and the distance is the height of the well.
The weight of the bucket is given as 4 lb, and the weight of the water is given as 44 lb. So, the total weight is 4 lb (bucket) + 44 lb (water) = 48 lb.
The distance to be lifted is 70 ft.
To approximate the required work by a Riemann sum, we can break down the distance into small intervals and calculate the work done at each interval using the given rates.
Let's choose a small interval Δx for the Riemann sum. The width of each interval is Δx = 2 ft (given rate of 2 ft/s).
For each interval, the work done is the force (weight) multiplied by the distance (Δx).
At each interval, the weight of the bucket and the water decreases due to the water leaking out. The rate of water leakage is given as 0.2 lb/s.
To calculate the work done at each interval, we need to find the weight at each interval.
Starting from the bottom of the well, the weight of the bucket and water at each interval can be calculated as:
Weight = 48 lb - 0.2 lb/s * t
Where t is the time in seconds since the bucket started being pulled.
Now, we can calculate the work done at each interval by multiplying the weight with Δx.
Work at each interval = Weight * Δx
= (48 lb - 0.2 lb/s * t) * 2 ft
To approximate the required work, we need to sum up the work done at each interval using a Riemann sum.
Riemann sum ≈ ∑[(48 lb - 0.2 lb/s * t) * 2 ft]
To calculate the exact value of the Riemann sum, we would need to know the time it takes to pull the bucket to the top of the well. However, without that information, we can only provide an approximation using this Riemann sum formula.