Question

find the length to three significant digits of the arc intercepted by a central angle θ in a circle of radius r.
r=18.7 cm, θ=\frac{2π}{9} radians

the length of the intercepted arc is approximately \square cm
(round to one decimal place as needed.)

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc \( s \) intercepted by a central angle \( \theta \) (in radians) in a circle of radius \( r \) is \( s = r\theta \).

Step2: Substitute given values

We are given \( r = 18.7 \) cm and \( \theta=\frac{2\pi}{9} \) radians. Substitute these values into the formula:
\( s = 18.7\times\frac{2\pi}{9} \)
First, calculate \( 18.7\times2 = 37.4 \), so \( s=\frac{37.4\pi}{9} \)
Now, use \( \pi\approx3.1416 \) to approximate:
\( s\approx\frac{37.4\times3.1416}{9} \)
Calculate \( 37.4\times3.1416\approx37.4\times3.1416 = 117.49584 \)
Then, divide by 9: \( s\approx\frac{117.49584}{9}\approx13.055 \)

Step3: Round to three significant digits

Three significant digits of \( 13.055 \) is \( 13.1 \) (since the fourth digit is 5, we round up the third digit).

Answer:

\( 13.1 \)