Question

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each arc is shown in blue. each indicated angle is the central angle that intercepts the given arc.

1. given: ( r = 4 ) inches and ( overline{ab}=\frac{8pi}{3} ) inches 18. given: ( r = 18 mathrm{~cm} ) and ( overline{ge}=21pi mathrm{~cm} )

find ( mangle bca ) in radians. find ( mangle gfe ) in radians.

1. given: ( r = 10 mathrm{~m} ) and ( overline{nl}=\frac{15pi}{2} mathrm{~m} ) 19. given: ( r = 27 mathrm{yd} ) and ( overline{pq}=3pi mathrm{yd} )

find ( mangle lmn ) in radians. find ( mangle prq ) in radians.

1. each radius and arc length includes a unit such as feet or meters. explain why radian measures do not include a unit.

Explanation:

Step1: Use the arc - length formula

The formula for arc - length is \(s = r\theta\), where \(s\) is the arc - length, \(r\) is the radius, and \(\theta\) is the central angle in radians. So, \(\theta=\frac{s}{r}\)

Step2: Solve for \(\theta\) in each problem

• Problem 16:

Given \(r = 4\) inches and \(s=\widehat{AB}=\frac{8\pi}{3}\) inches.
Substitute into \(\theta=\frac{s}{r}\), we get \(\theta=\frac{\frac{8\pi}{3}}{4}=\frac{2\pi}{3}\) radians.

• Problem 17:

Given \(r = 10\) m and \(s=\widehat{NL}=\frac{15\pi}{2}\) m.
Substitute into \(\theta=\frac{s}{r}\), we get \(\theta=\frac{\frac{15\pi}{2}}{10}=\frac{3\pi}{4}\) radians.

• Problem 18:

Given \(r = 18\) cm and \(s=\widehat{GE}=21\pi\) cm.
Substitute into \(\theta=\frac{s}{r}\), we get \(\theta=\frac{21\pi}{18}=\frac{7\pi}{6}\) radians.

• Problem 19:

Given \(r = 27\) yd and \(s=\widehat{PQ}=3\pi\) yd.
Substitute into \(\theta=\frac{s}{r}\), we get \(\theta=\frac{3\pi}{27}=\frac{\pi}{9}\) radians.

Step3: Explain why radian measures have no unit

Radian is defined as the ratio of the arc - length \(s\) to the radius \(r\) (\(\theta=\frac{s}{r}\)). Since both \(s\) and \(r\) have the same unit (e.g., meters, inches, etc.), when we take the ratio \(\frac{s}{r}\), the units cancel out. For example, if \(s\) is in meters and \(r\) is in meters, \(\frac{\text{meters}}{\text{meters}} = 1\) (a dimensionless quantity)

Answer:

1. \(\frac{2\pi}{3}\) radians
2. \(\frac{3\pi}{4}\) radians
3. \(\frac{7\pi}{6}\) radians
4. \(\frac{\pi}{9}\) radians
5. Radian is the ratio of arc - length to radius (\(\theta=\frac{s}{r}\)), and units of \(s\) and \(r\) cancel out.