Question

In circle R with m, angle, Q, R, S, equals, 60, degreesm∠QRS=60 ∘ and Q, R, equals, 14QR=14 units, find the length of arc, Q, S QS ⌢​. Round to the nearest hundredth.

Answer 1

To find the length of the arc \( QS \) in circle \( R \) given that the measure of angle \( \angle QRS = 60^\circ \) and the radius \( QR = 14 \) units, we can use the formula for the length of an arc.

The formula for the length of an arc \( L \) in a circle is given by:

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

where \( \theta \) is the central angle in degrees and \( C \) is the circumference of the circle. The circumference \( C \) of a circle is calculated using the formula:

\[
C = 2\pi r
\]

where \( r \) is the radius.

Here, since \( QR \) is the radius, we have \( r = 14 \) units. Let's calculate the circumference first:

\[
C = 2\pi \cdot 14 = 28\pi
\]

Now, we can calculate the length of the arc \( QS \):

1. Convert the angle to a fraction of the circle:

\[
L = \frac{60}{360} \times 28\pi
\]

2. Simplify \( \frac{60}{360} \):

\[
\frac{60}{360} = \frac{1}{6}
\]

3. Substitute back into the formula for the arc length:

\[
L = \frac{1}{6} \times 28\pi = \frac{28\pi}{6} = \frac{14\pi}{3}
\]

4. Now, calculate this value numerically. Using \( \pi \approx 3.14159\):

\[
L \approx \frac{14 \times 3.14159}{3} \approx \frac{43.98226}{3} \approx 14.66075
\]

Finally, rounding to the nearest hundredth, the length of arc \( QS \) is approximately:

\[
\boxed{14.66}
\] units.