Final answer:
The angle measure of an arc 2π feet long in a circle with a radius of 8 feet is 45 degrees, calculated by the proportion of the arc length to the circumference related to the full 360-degree circle.
Step-by-step explanation:
The student has asked to determine the angle measure of an arc that is 2π feet long in a circle with a radius of 8 feet. The circumference of a circle is 2πr, and this circumference corresponds to a 360-degree angle in the circle. Given that the arc length for a full revolution (360 degrees) is the circumference (2πr), we can find the measure of the angle for any arc length using the proportion:
Δθ = (arc length / circumference) × 360°
Substituting the given values (arc length = 2π feet and r = 8 feet), we use the formula:
Δθ = (2π / (2π × 8)) × 360°
Simplifying the expression:
Δθ = (1/8) × 360°
Δθ = 45
Therefore, the angle measure of an arc 2π feet long in a circle with a radius of 8 feet is 45 degrees.