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