Find the length of an arc of a circle of radius 75 cm that spans a central angle of measure `126^(@)`

Answer

To find the length of an arc of a circle given the radius and the central angle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values**:
- Radius (r) = 75 cm
- Central angle (θ) = 126 degrees

2. **Convert the angle from degrees to radians**:
The formula to convert degrees to radians is:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
Applying this to our angle:
\[ θ = 126 \times \frac{\pi}{180} = \frac{126\pi}{180} \]
Simplifying this fraction:
\[ θ = \frac{7\pi}{10} \text{ radians} \]

3. **Use the formula for the length of an arc**:
The formula for the length of an arc (l) is given by:
\[ l = r \times θ \]
Substituting the values we have:
\[ l = 75 \times \frac{7\pi}{10} \]

4. **Calculate the length of the arc**:
First, calculate:
\[ l = 75 \times \frac{7\pi}{10} = \frac{525\pi}{10} = 52.5\pi \]
Now, using the approximate value of \(\pi \approx 3.14\):
\[ l \approx 52.5 \times 3.14 \approx 164.85 \text{ cm} \]

5. **Round the answer to the nearest whole number**:
Rounding 164.85 gives us:
\[ l \approx 165 \text{ cm} \]

### Final Answer:
The length of the arc is approximately **165 cm**.