Question

the radius of a circle is 9 inches. what is the length of a 102° arc?

r=9 in

102°

give the exact answer in simplest form.

inches

π

submit

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc is \( L=\frac{\theta}{360^\circ}\times2\pi r \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.

Step2: Substitute given values

Here, \( \theta = 102^\circ \) and \( r = 9 \) inches. Substitute these into the formula:
\( L=\frac{102^\circ}{360^\circ}\times2\pi\times9 \)

Step3: Simplify the expression

First, simplify \( \frac{102}{360}\times2\times9 \). \( 2\times9 = 18 \), then \( \frac{102}{360}\times18=\frac{102\times18}{360}=\frac{102}{20}=\frac{51}{10} \). So \( L=\frac{51}{10}\pi\times3 \)? Wait, no, wait: \( \frac{102}{360}\times2\pi\times9=\frac{102\times2\times9\pi}{360}=\frac{102\times18\pi}{360}=\frac{102\pi}{20}=\frac{51\pi}{10}\times3 \)? No, wait, 18/360 = 1/20, so 102×1/20×π? Wait, no, 2×9 = 18, 18/360 = 1/20, so 102×1/20×π? Wait, no, 102/360 × 18π = (102×18π)/360. Simplify 18/360 = 1/20, so 102×π/20 = 51π/10? Wait, 102 divided by 2 is 51, 18 divided by 2 is 9, 360 divided by 2 is 180. So (51×9π)/180 = (459π)/180 = 51π/20? Wait, no, let's do it step by step. 102/360 = 17/60 (dividing numerator and denominator by 6). Then 2πr is the circumference, which is 2π×9 = 18π. Then arc length is (17/60)×18π. 18 and 60 have a common factor of 6: 18÷6 = 3, 60÷6 = 10. So (17×3π)/10 = 51π/10? Wait, no, 17×3 = 51, so 51π/10? Wait, 102/360 = 17/60, 17/60 × 18π = (17×18π)/60 = (17×3π)/10 = 51π/10. Yes, that's correct. So the arc length is \( \frac{51}{10}\pi \) or \( 5.1\pi \), but in simplest form, \( \frac{51\pi}{10} \) can be written as \( \frac{51}{10}\pi \) or simplified further? Wait, 51 and 10 have no common factors, so \( \frac{51\pi}{10} \) is the exact form. Wait, wait, let's recalculate:

\( \frac{102}{360} \times 2\pi \times 9 \)

First, 2×9 = 18. So:

\( \frac{102}{360} \times 18\pi \)

Simplify \( \frac{102}{360} \times 18 \):

18/360 = 1/20, so 102×1/20 = 102/20 = 51/10. So:

\( \frac{51}{10}\pi \times 1 \)? Wait, no, 18π times 102/360 is (102×18π)/360. Let's divide numerator and denominator by 18: 102/20 π = 51/10 π. Yes, that's correct. So the arc length is \( \frac{51}{10}\pi \) inches, which is \( 5.1\pi \) inches, but in exact form, it's \( \frac{51\pi}{10} \).

Wait, but let's check the formula again. The arc length is a fraction of the circumference, where the fraction is the central angle over 360 degrees. So circumference is \( 2\pi r = 18\pi \). The central angle is 102 degrees, so the fraction is 102/360. So arc length is (102/360)18π. 102 divided by 360 is 17/60, 17/60 18π = (1718π)/60 = (173π)/10 = 51π/10. Yes, that's correct. So the exact length is \( \frac{51\pi}{10} \) inches.

Answer:

\( \frac{51}{10}\pi \) (or \( 5.1\pi \), but as an exact form, \( \frac{51\pi}{10} \))