Final answer:
To find the radius of a circle with a given arc length of 57 units and an arc measure of 18 degrees, one uses the arc length formula. After setting up the equation 57 = 18/360 × 2πr and simplifying, the radius is found to be approximately 57.9 units.
Step-by-step explanation:
To find the radius of a circle when given the measure of an arc (in degrees) and the length of the arc (in units), we can use the formula for the arc length of a circle, which is θ/360 × 2πr, where θ is the angle in degrees and r is the radius. We are given that the measure of the arc θ is 18 degrees and the arc length is 57 units.
First, we set up the equation:
57 = 18/360 × 2πr
Next, we simplify and solve for r:
57 = 18/360 × 2πr
57 = 1/20 × 2πr (because 18/360 simplifies to 1/20)
57 = π/10 × r (2π divided by 20 is π/10)
57 × 10/π = r (multiplying both sides by 10/π to isolate r)
r = 57 × 10/π
r ≈ 181.9/π
r ≈ 57.9 (using π ≈ 3.14159)
Therefore, the radius of the circle is approximately 57.9 units.