To find the length of an arc with a given angle in degrees, you can use the formula for the circumference, \(\( C \)\), of a circle and then find the fraction of the circumference that corresponds to the angle of the arc.
The formula for the circumference of a circle is \(\( C = 2\pi r \)\), where \(\( r \)\) is the radius.
Given that the radius, \(\( r \)\), of the circle is 3 miles, and we want to find the arc length for an angle of 75 degrees, we can do the following steps:
1. Calculate the circumference of the full circle by using the formula \(\( C = 2\pi r \)\). In this case, we have:
\(\[ C_{\text{full}} = 2\pi \times 3 \]\)
\(\[ C_{\text{full}} = 6\pi \, \text{miles} \]\)
2. The length of the arc will be a fraction of the full circumference, given by the ratio of the angle of the arc (75 degrees) to the full angle of a circle (360 degrees). So the fraction of the circle that the arc represents is:
\(\[ \text{Arc fraction} = \frac{\text{Angle of arc}}{\text{Full angle of circle}} = \frac{75}{360} \]\)
3. Finally, calculate the arc length by multiplying the full circumference by the arc fraction to find the length of the arc:
\(\[ \text{Arc length} = C_{\text{full}} \times \text{Arc fraction} \]\)
\(\[ \text{Arc length} = 6\pi \times \frac{75}{360} \]\)
\(\[ \text{Arc length} = \frac{6\pi \times 75}{360} \]\)
\(\[ \text{Arc length} = \frac{\pi \times 75}{60} \]\)
\(\[ \text{Arc length} = \frac{5\pi}{8} \, \text{miles} \]\)
Thus, the exact length of the 75° arc of a circle with a radius of 3 miles, in simplest form, is \(\( \frac{5\pi}{8} \)\) miles.
