Refer to the figure to the right.
​(a) How many inches will the weight in the figure rise if the pulley is rotated through an angle of 73degrees 40​'?
​(b)  Through what​ angle, to the nearest​ minute, must the pulley be rotated to raise the weight 5 ​in.?
9.82 in
Question content area bottom
Part 1
​(a) The weight in the figure will rise 
  
12.6 inches.
​(Do not round until the final answer. Then round to the nearest tenth as​ needed.)
Part 2
​(b) In order for the weight to rise 5 ​inches, the pulley must rotate through 
  
29degrees 
  
12​'.
​(Do not round until the final answer. Then round to the nearest minute as​ needed.)

### Part 1 (a):

To determine how many inches the weight will rise when the pulley is rotated through an angle of 73°40', we use the relationship between the rotation of the pulley and the vertical displacement of the weight.

Assume the pulley has a radius \( r = 9.82 \, \text{inches} \). When the pulley rotates, the weight rises according to the arc length of the rotation, which can be calculated using the formula:

\[
\text{Arc length} = r \times \theta
\]

Where:
- \( r \) is the radius of the pulley (9.82 inches),
- \( \theta \) is the angle of rotation in radians.

First, we need to convert the angle \( \theta \) from degrees and minutes to radians. We know that:

\[
1^\circ = \frac{\pi}{180} \text{ radians}
\]

Given the angle \( 73^\circ 40' \), we first convert the minutes to a fraction of a degree:

\[
40' = \frac{40}{60}^\circ = \frac{2}{3}^\circ
\]

Thus, the total angle \( \theta \) is:

\[
\theta = 73^\circ + \frac{2}{3}^\circ = 73.6667^\circ
\]

Now, convert the angle to radians:

\[
\theta = 73.6667 \times \frac{\pi}{180} \approx 1.2856 \, \text{radians}
\]

Now, calculate the arc length:

\[
\text{Arc length} = 9.82 \times 1.2856 \approx 12.6 \, \text{inches}
\]

So, the weight will rise **12.6 inches**.

### Part 2 (b):

Now, to find the angle through which the pulley must rotate to raise the weight by 5 inches, we use the same relationship. The formula for arc length is:

\[
\text{Arc length} = r \times \theta
\]

Where:
- The arc length (displacement) is 5 inches,
- The radius \( r = 9.82 \, \text{inches} \),
- \( \theta \) is the angle in radians.

Rearranging the formula to solve for \( \theta \):

\[
\theta = \frac{\text{Arc length}}{r}
\]

Substitute the known values:

\[
\theta = \frac{5}{9.82} \approx 0.5097 \, \text{radians}
\]

Now, convert \( \theta \) from radians back to degrees:

\[
\theta = 0.5097 \times \frac{180}{\pi} \approx 29.2^\circ
\]

To convert the decimal part to minutes, multiply the decimal by 60:

\[
0.2 \times 60 = 12' \quad (\text{12 minutes})
\]

Thus, the pulley must be rotated through **29°12'** to raise the weight by 5 inches.

---

### Summary:
- **(a)** The weight will rise **12.6 inches** when the pulley rotates through 73°40'.
- **(b)** The pulley must rotate through **29°12'** to raise the weight by 5 inches.

Would you like to see more details or have any further questions?

---

Here are some related questions:
1. How would the arc length change if the radius of the pulley were doubled?
2. If the angle of rotation was 90° instead of 73°40', how would the weight's displacement change?
3. How would the displacement change if the radius of the pulley was reduced by half?
4. How does the relationship between the arc length and the angle of rotation apply to a real-world situation, like lifting a load with a pulley?
5. What are the practical limits on how far a pulley can rotate without damaging the system?

**Tip:** Always check your angle units (degrees or radians) before applying formulas to avoid conversion errors.

Refer to the figure to the right.
(a) How many inches will the weight in the figure rise if the pulley is rotated through an angle of 73°40'?
(b) Through what angle, to the nearest minute, must the pulley be rotated to raise the weight 5 inches?
Radius: 9.82 in.

Learn how to determine the vertical displacement of a weight when a pulley rotates. This problem explores the relationship between arc length and rotation, using a pulley with a 9.82-inch radius. Calculate the height increase for a 73°40' rotation and determine the required rotation to raise the weight by 5 inches. Suitable for high school students studying geometry and trigonometry.

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