To find the length of the arc of a circle, you can use the formula:
\[ \text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r \]
where:
- \(\theta\) is the angle in degrees,
- \(r\) is the radius of the circle.
Given:
- The radius \(r = 18\) cm,
- The angle \(\theta = 75^\circ\).
Now, plug in the values:
\[ \text{Arc Length} = \frac{75}{360} \times 2\pi \times 18 \]
First, simplify the fraction:
\[ \frac{75}{360} = \frac{5}{24} \]
Now calculate the arc length:
\[ \text{Arc Length} = \frac{5}{24} \times 2 \times \pi \times 18 \]
Calculate \(2 \times 18 = 36\):
\[ \text{Arc Length} = \frac{5}{24} \times 36\pi \]
Now simplify:
\[ \text{Arc Length} = \frac{5 \times 36\pi}{24} \]
Divide \(36\) by \(24\):
\[ \frac{36}{24} = 1.5 \]
Thus, the expression becomes:
\[ \text{Arc Length} = 5 \times 1.5\pi = 7.5\pi \]
Finally, use \(\pi \approx 3.14\) to calculate a numerical approximation:
\[ \text{Arc Length} \approx 7.5 \times 3.14 \approx 23.49 , \text{cm} \]
So, the length of the arc is approximately \(23.49\) cm.