Question

how long is the arc intersected by a \\( \frac { 2 \pi } { 3 } \\) radian central angle in a circle with radius 7 feet?
\\( \frac { 2 \pi } { 21 } \\) feet \\( \frac { 7 \pi } { 3 } \\) feet \\( \frac { 21 } { 2 \pi } \\) feet \\( \frac { 14 \pi } { 3 } \\) feet
use the circle.
what is the radian measure for the angle to the
nearest hundredth? use 3.14 for \\( \pi \\).
\\( \sqrt { 2.36 } \\) radians
given a radius of 9 inches, estimate the length of
arc \\( s \\) to the nearest hundredth.
\\( \square \\) inches

Explanation:

First Question (Arc Length with Radius 7 ft and Central Angle \( \frac{2\pi}{3} \) Radians)

Step1: Recall the arc length formula

The formula for the length of an arc \( s \) is \( s = r\theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians.

Step2: Substitute the given values

Here, \( r = 7 \) feet and \( \theta=\frac{2\pi}{3} \). So, \( s = 7\times\frac{2\pi}{3}=\frac{14\pi}{3} \) feet.

Step1: Recall the degree to radian conversion formula

To convert degrees to radians, we use the formula \( \text{radians}=\text{degrees}\times\frac{\pi}{180} \).

Step2: Substitute the given degree measure

For \( 135^\circ \), we have \( 135\times\frac{\pi}{180}=\frac{3\pi}{4} \approx\frac{3\times3.14}{4}=\frac{9.42}{4} = 2.355\approx2.36 \) radians (which matches the given answer).

Step1: Recall the arc length formula

Using \( s = r\theta \), where \( r = 9 \) inches and \( \theta=\frac{3\pi}{4} \) (or we can use the degree - radian conversion in the formula \( s=r\times(\text{degrees}\times\frac{\pi}{180}) \)).

Step2: Calculate the arc length

Using \( s = r\theta \) with \( \theta=\frac{3\pi}{4} \) and \( r = 9 \), we get \( s=9\times\frac{3\pi}{4}=\frac{27\pi}{4}\approx\frac{27\times3.14}{4}=\frac{84.78}{4}=21.195\approx21.20 \) inches. Or using the degree formula: \( s = 9\times135\times\frac{\pi}{180}=9\times\frac{3\pi}{4}=\frac{27\pi}{4}\approx21.20 \) inches.

Answer:

\(\frac{14\pi}{3}\) feet

Second Question (Radian Measure of \( 135^\circ \))