To find the length of the arc formed by a central angle in a circle, you can use the formula:
\[ \text{Arc Length} = \frac{\theta}{360} \times 2\pi r \]
where:
- \( \theta \) is the central angle in degrees,
- \( r \) is the radius of the circle.
Given that the central angle \( \theta = 85\) degrees and the radius \( r = 16\) millimeters, we can substitute these values into the formula:
\[ \text{Arc Length} = \frac{85}{360} \times 2\pi \times 16 \]
First, simplify \( \frac{85}{360} \):
\[ \frac{85}{360} = \frac{17}{72} \]
Now, substitute this back into the formula for arc length:
\[ \text{Arc Length} = \frac{17}{72} \times 2\pi \times 16 \]
Calculating \( 2\pi \times 16 \):
\[ 2\pi \times 16 = 32\pi \]
Now, substituting this into the equation gives:
\[ \text{Arc Length} = \frac{17}{72} \times 32\pi \]
Now simplify \( \frac{17 \times 32\pi}{72} \):
\[ \text{Arc Length} = \frac{544\pi}{72} \]
Now simplify \( \frac{544}{72} \):
Dividing both the numerator and denominator by 8:
\[ \frac{544 \div 8}{72 \div 8} = \frac{68}{9} \]
Therefore, the length of the arc is:
\[ \text{Arc Length} = \frac{68\pi}{9} \text{ millimeters} \]
Thus, the final answer for the length of the arc, in terms of \(\pi\), is:
\[ \frac{68\pi}{9} \text{ millimeters} \]
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