Answer:
The relationship between the length of an arc, the radius of the circle, and the measure of the central angle that intercepts the arc is given by the formula:
length of arc = (central angle measure / 360°) x 2πr
We can use this formula to find the measure of the central angle, given the radius and the length of the intercepted arc. Substituting r = 6 and length of arc = 47 into the formula, we get:
47 = (central angle measure / 360°) x 2π(6)
Simplifying, we get:
47 = (central angle measure / 360°) x 12π
Dividing both sides by 12π and multiplying by 360°, we get:
central angle measure = (47 / 12π) x 360°
Using a calculator to approximate π to three decimal places, we get:
central angle measure ≈ 56.67°
Therefore, the measure of the central angle that intercepts an arc of length 47 on a circle with radius 6 is approximately 56.67 degrees.