the pendulum on a clock swings through an angle `25^(@)`, and the tip sweeps out an arc of 12 inches. How long is the pendulum?

Options

A. 1.67 inches
B. 13.8 inches
C. 27.5 inches
D. 43.2 inches

Explanation

To find the length of the pendulum, we can use the relationship between the arc length, the radius (length of the pendulum), and the angle in radians. Here’s a step-by-step solution:

### Step 1: Understand the relationship
The formula that relates the arc length (s), the radius (r), and the angle in radians (θ) is given by:
\[ s = r \cdot \theta \]
Where:
- \( s \) is the arc length (12 inches in this case),
- \( r \) is the length of the pendulum (which we need to find),
- \( \theta \) is the angle in radians.

### Step 2: Convert degrees to radians
We need to convert the angle from degrees to radians. The conversion formula is:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
For our case:
\[ \theta = 25^\circ \times \frac{\pi}{180} = \frac{25\pi}{180} \]

### Step 3: Simplify the angle in radians
We can simplify \( \frac{25\pi}{180} \):
\[ \theta = \frac{25\pi}{180} = \frac{5\pi}{36} \]

### Step 4: Substitute values into the formula
Now we can substitute the values into the arc length formula:
\[ s = r \cdot \theta \]
Substituting \( s = 12 \) inches and \( \theta = \frac{5\pi}{36} \):
\[ 12 = r \cdot \frac{5\pi}{36} \]

### Step 5: Solve for the length of the pendulum (r)
To find \( r \), we rearrange the equation:
\[ r = \frac{12 \cdot 36}{5\pi} \]

### Step 6: Calculate the value
Now we can calculate the value:
\[ r = \frac{432}{5\pi} \]
Using \( \pi \approx 3.14 \):
\[ r \approx \frac{432}{5 \times 3.14} \approx \frac{432}{15.7} \approx 27.5 \text{ inches} \]

### Conclusion
The length of the pendulum is approximately **27.5 inches**.