To determine the length of an arc in a circle given its radius and the angle in radians, you can use the arc length formula:
[tex]\[ \text{Arc Length} = \text{Radius} \times \text{Angle in Radians} \][/tex]
Here’s a detailed step-by-step solution for calculating the arc length:
1. Identify the given values:
- Radius ([tex]\(r\)[/tex]) = 102 miles
- Angle ([tex]\(\theta\)[/tex]) = 4.7 radians
2. Substitute the given values into the formula:
[tex]\[ \text{Arc Length} = 102 \, \text{miles} \times 4.7 \, \text{radians} \][/tex]
3. Calculate the arc length:
[tex]\[ \text{Arc Length} = 102 \times 4.7 = 479.4 \, \text{miles} \][/tex]
4. Round the result to the nearest tenth:
The calculated arc length is already 479.4 miles, which is precisely to the nearest tenth.
So, the length of the arc associated with an angle of 4.7 radians in a circle with a radius of 102 miles is [tex]\( \mathbf{479.4 \, \text{miles}} \)[/tex].
