Question

If a circle has a central angle of 85° and a radius of 16 millimeters, how long is the arc formed by that angle? Leave your answer in terms of π. (1 point) Responses 64π mm 64π mm 68π9 mm 68π9 mm 68π18 mm 68π18 mm 544π9 mm

Answer 1

To find the length of the arc formed by a central angle in a circle, you can use the formula:

\[
L = \frac{\theta}{360} \times C
\]

where \(L\) is the arc length, \(\theta\) is the central angle in degrees, and \(C\) is the circumference of the circle.

1. First, calculate the circumference \(C\) of the circle using the radius \(r = 16\) mm:

\[
C = 2\pi r = 2\pi \times 16 = 32\pi \text{ mm}
\]

2. Now, use the central angle \(\theta = 85^\circ\) to find the arc length:

\[
L = \frac{85}{360} \times 32\pi
\]

3. Simplify the fraction:

\[
\frac{85}{360} = \frac{17}{72}
\]

4. Substitute back into the arc length formula:

\[
L = \frac{17}{72} \times 32\pi
\]

5. Multiply:

\[
L = \frac{17 \times 32\pi}{72} = \frac{544\pi}{72}
\]

6. Simplify the fraction:

\[
\frac{544\pi}{72} = \frac{68\pi}{9} \text{ mm}
\]

Thus, the length of the arc is:

\[
\boxed{\frac{68\pi}{9} \text{ mm}}
\]