Answer :
To solve this problem, we need to find the measure of the central angle whose radii define an arc on a circle. We are given the radius of the circle and the length of the arc.
Here are the step-by-step instructions and reasoning:
1. Understand the Relationship Between Arc Length and Central Angle:
The arc length [tex]\( L \)[/tex] of a circle is related to the radius [tex]\( r \)[/tex] and the central angle [tex]\( \theta \)[/tex] (in radians) by the formula:
[tex]\[ L = r \times \theta \][/tex]
2. Given Values:
- Radius [tex]\( r = 3 \)[/tex] centimeters
- Arc length [tex]\( L = 4 \)[/tex] centimeters
3. Rewrite the Formula to Solve for the Central Angle [tex]\( \theta \)[/tex]:
To find the central angle [tex]\( \theta \)[/tex] in radians, we rearrange the formula:
[tex]\[ \theta = \frac{L}{r} \][/tex]
4. Substitute the Given Values:
[tex]\[ \theta = \frac{4 \text{ cm}}{3 \text{ cm}} = \frac{4}{3} \text{ radians} \][/tex]
5. Convert the Central Angle from Radians to Degrees:
To convert from radians to degrees, use the conversion factor [tex]\( 180^\circ = \pi \)[/tex] radians. Thus, the conversion formula is:
[tex]\[ \theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180^\circ}{\pi} \][/tex]
Now, substitute [tex]\( \theta_{\text{radians}} = \frac{4}{3} \)[/tex]:
[tex]\[ \theta_{\text{degrees}} = \frac{4}{3} \times \frac{180^\circ}{\pi} \][/tex]
6. Calculate the Degrees:
[tex]\[ \theta_{\text{degrees}} = \frac{4}{3} \times \frac{180^\circ}{\pi} = \frac{720^\circ}{3\pi} = \frac{240^\circ}{\pi} \][/tex]
Using the approximation [tex]\( \pi \approx 3.1416 \)[/tex]:
[tex]\[ \theta_{\text{degrees}} \approx \frac{240^\circ}{3.1416} \approx 76.39^\circ \][/tex]
So, the measure of the central angle in radians is [tex]\( \frac{4}{3} \)[/tex] radians, and in degrees, it is approximately [tex]\( 76.39^\circ \)[/tex].
Summary:
- Central angle in radians: [tex]\( \frac{4}{3} \)[/tex] radians
- Central angle in degrees: [tex]\( 76.39^\circ \)[/tex]
Here are the step-by-step instructions and reasoning:
1. Understand the Relationship Between Arc Length and Central Angle:
The arc length [tex]\( L \)[/tex] of a circle is related to the radius [tex]\( r \)[/tex] and the central angle [tex]\( \theta \)[/tex] (in radians) by the formula:
[tex]\[ L = r \times \theta \][/tex]
2. Given Values:
- Radius [tex]\( r = 3 \)[/tex] centimeters
- Arc length [tex]\( L = 4 \)[/tex] centimeters
3. Rewrite the Formula to Solve for the Central Angle [tex]\( \theta \)[/tex]:
To find the central angle [tex]\( \theta \)[/tex] in radians, we rearrange the formula:
[tex]\[ \theta = \frac{L}{r} \][/tex]
4. Substitute the Given Values:
[tex]\[ \theta = \frac{4 \text{ cm}}{3 \text{ cm}} = \frac{4}{3} \text{ radians} \][/tex]
5. Convert the Central Angle from Radians to Degrees:
To convert from radians to degrees, use the conversion factor [tex]\( 180^\circ = \pi \)[/tex] radians. Thus, the conversion formula is:
[tex]\[ \theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180^\circ}{\pi} \][/tex]
Now, substitute [tex]\( \theta_{\text{radians}} = \frac{4}{3} \)[/tex]:
[tex]\[ \theta_{\text{degrees}} = \frac{4}{3} \times \frac{180^\circ}{\pi} \][/tex]
6. Calculate the Degrees:
[tex]\[ \theta_{\text{degrees}} = \frac{4}{3} \times \frac{180^\circ}{\pi} = \frac{720^\circ}{3\pi} = \frac{240^\circ}{\pi} \][/tex]
Using the approximation [tex]\( \pi \approx 3.1416 \)[/tex]:
[tex]\[ \theta_{\text{degrees}} \approx \frac{240^\circ}{3.1416} \approx 76.39^\circ \][/tex]
So, the measure of the central angle in radians is [tex]\( \frac{4}{3} \)[/tex] radians, and in degrees, it is approximately [tex]\( 76.39^\circ \)[/tex].
Summary:
- Central angle in radians: [tex]\( \frac{4}{3} \)[/tex] radians
- Central angle in degrees: [tex]\( 76.39^\circ \)[/tex]
