To calculate the work required to lift the bag, we need to determine the total weight of the bag and rope at each height and then sum up the work done at each height.
Let's break down the problem step by step:
1. The bag weighs 30 pounds, and for each foot it is raised, it loses 0.25 pounds due to the leakage. So at height h feet, the weight of the bag is (30 - 0.25h) pounds.
2. The rope weighs 2 pounds per foot. At height h feet, the weight of the rope is 2h pounds.
3. The total weight at height h is the sum of the weight of the bag and the rope:
Weight at height h = (30 - 0.25h) + 2h = 30 + 1.75h pounds.
4. The work done to lift a weight W through a distance d is given by the formula:
Work = Force × Distance.
5. In this case, the force is equal to the weight, and the distance is the height. So the work done at height h is:
Work at height h = (30 + 1.75h) × h = 30h + 1.75h^2.
6. To find the total work required to lift the bag to the top of the 80-foot building, we need to sum up the work done at each height from 0 to 80:
Total work = ∑(30h + 1.75h^2) for h = 0 to 80.
To calculate this sum, we can use the formula for the sum of the first n squares:
∑(h^2) = (n/6) × (n+1) × (2n+1).
Plugging in the values, we get:
Total work = ∑(30h + 1.75h^2) for h = 0 to 80
= 30 × ∑(h) + 1.75 × ∑(h^2) for h = 0 to 80
= 30 × (80 × 81 / 2) + 1.75 × (80/6) × (80+1) × (2×80+1).
Evaluating this expression will give us the total work required to lift the bag.