Question

In circle S with m, angle, R, S, T, equals, 30, degreesm∠RST=30
∘
and R, S, equals, 17RS=17 units, find the length of arc, R, T
RT
⌢
. Round to the nearest hundredth.

Answer 1

To find the length of arc \( RT \) in circle \( S \) given that \( m \angle RST = 30^\circ \) and the radius \( RS = 17 \) units, we can use the formula for the length of an arc:

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

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

First, we need to calculate the circumference \( C \) of the circle. The circumference is given by the formula:

\[
C = 2\pi r
\]

where \( r \) is the radius.

Here, the radius \( r = 17 \) units. So,

\[
C = 2\pi(17) = 34\pi
\]

Now, we substitute \( \theta = 30^\circ \) into the arc length formula:

\[
L = \frac{30}{360} \times (34\pi)
\]

This simplifies to:

\[
L = \frac{1}{12} \times 34\pi
\]

Calculating this gives:

\[
L = \frac{34\pi}{12} = \frac{17\pi}{6}
\]

Now, to find the numerical value, we can use the approximation \( \pi \approx 3.14 \):

\[
L \approx \frac{17 \times 3.14}{6} \approx \frac{53.38}{6} \approx 8.896666
\]

Rounding this to the nearest hundredth, we find:

\[
L \approx 8.90
\]

Thus, the length of arc \( RT \) is approximately \(\boxed{8.90}\) units.