A heavy rope, 30 ft long, weighs 0.4 lb/ft and hangs over the edge of a building 100 ft high. How much work *W* is done in pulling the rope to the top of the building?

(a)

How much work *W* is done in pulling the rope to the top of the building? Show how to approximate the required work by a Riemann sum, then express the work as an integral and evaluate it.

(b)

A heavy rope, 30 ft long, weighs 0.4 lb/ft and hangs over the edge of a building 100 ft high. How much work *W* is done in pulling half the rope to the top of the building?

### Problem Overview:

We need to calculate the amount of work done in pulling a heavy rope over the edge of a building. Work is defined as force multiplied by distance. Since the force in this case is the weight of the rope, and the length of the rope being lifted varies as more rope is pulled to the top, the problem is best approached using integration.

### Part (a): **Pulling the Entire Rope to the Top**

**Step 1: Set Up the Problem**

- The total length of the rope is 30 feet.
- The rope weighs 0.4 lb/ft, so the total weight of the rope is \( 30 \times 0.4 = 12 \, \text{lb} \).
- We need to calculate the work required to pull the entire 30-foot rope to the top.

Let \( x \) represent the length of the rope that has already been pulled up. At any moment, the amount of rope hanging down is \( 30 - x \) feet.

The weight of the hanging portion of the rope at that moment is:
\[
\text{Weight of hanging rope} = 0.4 \times (30 - x) \, \text{lb}
\]

To pull this small portion \( dx \) of the rope up a distance of \( x \), the force applied is the weight of the hanging rope, and the work done to move this portion is:
\[
dW = \text{Force} \times \text{Distance} = 0.4(30 - x) \, dx
\]

**Step 2: Set Up the Integral**

To find the total work, integrate this expression from \( x = 0 \) (when none of the rope is pulled up) to \( x = 30 \) (when the entire rope is pulled up):
\[
W = \int_0^{30} 0.4(30 - x) \, dx
\]

**Step 3: Evaluate the Integral**

First, simplify the integrand:
\[
W = 0.4 \int_0^{30} (30 - x) \, dx
\]
Now, evaluate the integral:
\[
W = 0.4 \left[ 30x - \frac{x^2}{2} \right]_0^{30}
\]

Substitute the limits:
\[
W = 0.4 \left[ 30(30) - \frac{30^2}{2} \right]
\]
\[
W = 0.4 \left[ 900 - 450 \right] = 0.4 \times 450 = 180 \, \text{ft-lb}
\]

Thus, the total work required to pull the entire rope to the top of the building is **180 ft-lb**.

---

### Part (b): **Pulling Half the Rope to the Top**

In this part, we only pull half of the rope (15 feet) to the top of the building.

**Step 1: Set Up the Problem**

In this case, we are lifting only half the rope. So we integrate the same expression, but only over the interval \( x = 0 \) to \( x = 15 \).

The integral is:
\[
W = \int_0^{15} 0.4(30 - x) \, dx
\]

**Step 2: Evaluate the Integral**

Again, simplify the integrand:
\[
W = 0.4 \int_0^{15} (30 - x) \, dx
\]
Now evaluate the integral:
\[
W = 0.4 \left[ 30x - \frac{x^2}{2} \right]_0^{15}
\]

Substitute the limits:
\[
W = 0.4 \left[ 30(15) - \frac{15^2}{2} \right]
\]
\[
W = 0.4 \left[ 450 - 112.5 \right] = 0.4 \times 337.5 = 135 \, \text{ft-lb}
\]

Thus, the work required to pull half of the rope to the top of the building is **135 ft-lb**.

---

### Summary of Results:
1. The work required to pull the entire rope to the top of the building is **180 ft-lb**.
2. The work required to pull half the rope to the top is **135 ft-lb**.

---

Do you want further details or have any questions? Here are some related questions to consider:

1. How is the concept of force integrated over a varying distance to find work in more complex scenarios?
2. How would the problem change if the weight of the rope were not uniform along its length?
3. Can this method be applied to objects other than ropes? What conditions need to be met?
4. How does the length of the rope affect the total work required if we change the length of the hanging rope?
5. What physical implications would occur if we changed the gravitational force or applied it in a different direction?

**Tip:** Always ensure your force or weight functions are properly defined before setting up integrals for work.

A heavy rope, 30 ft long, weighs 0.4 lb/ft and hangs over the edge of a building 100 ft high. How much work W is done in pulling the rope to the top of the building?

(a) How much work W is done in pulling the rope to the top of the building? Show how to approximate the required work by a Riemann sum, then express the work as an integral and evaluate it.

(b) A heavy rope, 30 ft long, weighs 0.4 lb/ft and hangs over the edge of a building 100 ft high. How much work W is done in pulling half the rope to the top of the building?

Calculate the work required to pull a 30-ft heavy rope weighing 0.4 lb/ft over a 100-ft building. This problem covers calculus concepts such as integration and Riemann sums. It evaluates the total work to pull the entire rope and half the rope to the top, yielding solutions of 180 ft-lb and 135 ft-lb, respectively.

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