Answer :
Sure, let's go through the steps to solve this problem:
1. Understand the Problem:
- We have a circle with radius [tex]\(3\)[/tex] meters.
- The central angle [tex]\(XYZ\)[/tex] is [tex]\(70^\circ\)[/tex].
- We need to find the length of the minor arc [tex]\(XZ\)[/tex] and round it to the nearest tenth of a meter.
2. Convert the Central Angle to Radians:
- The formula to convert degrees to radians is: [tex]\(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)[/tex].
- Plug in the central angle in degrees:
[tex]\[ 70^\circ \times \frac{\pi}{180} = \frac{70\pi}{180} = \frac{7\pi}{18} \approx 1.2217 \text{ radians} (approximated to 4 decimal places) \][/tex]
3. Calculate the Arc Length:
- The formula to calculate the arc length [tex]\(L\)[/tex] of a circle is: [tex]\(L = r \cdot \theta\)[/tex], where [tex]\(r\)[/tex] is the radius and [tex]\(\theta\)[/tex] is the central angle in radians.
- Using the radius [tex]\(r = 3\)[/tex] meters and [tex]\(\theta \approx 1.2217\)[/tex] radians:
[tex]\[ L = 3 \times 1.2217 \approx 3.6652 \text{ meters} (approximated to 4 decimal places) \][/tex]
4. Round the Arc Length:
- Round the arc length to the nearest tenth of a meter:
[tex]\[ 3.6652 \approx 3.7 \text{ meters} \][/tex]
5. Conclusion:
- The approximate length of minor arc [tex]\(XZ\)[/tex] is [tex]\(3.7\)[/tex] meters.
So, the correct answer is:
[tex]\[ \boxed{3.7 \text{ meters}} \][/tex]
1. Understand the Problem:
- We have a circle with radius [tex]\(3\)[/tex] meters.
- The central angle [tex]\(XYZ\)[/tex] is [tex]\(70^\circ\)[/tex].
- We need to find the length of the minor arc [tex]\(XZ\)[/tex] and round it to the nearest tenth of a meter.
2. Convert the Central Angle to Radians:
- The formula to convert degrees to radians is: [tex]\(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)[/tex].
- Plug in the central angle in degrees:
[tex]\[ 70^\circ \times \frac{\pi}{180} = \frac{70\pi}{180} = \frac{7\pi}{18} \approx 1.2217 \text{ radians} (approximated to 4 decimal places) \][/tex]
3. Calculate the Arc Length:
- The formula to calculate the arc length [tex]\(L\)[/tex] of a circle is: [tex]\(L = r \cdot \theta\)[/tex], where [tex]\(r\)[/tex] is the radius and [tex]\(\theta\)[/tex] is the central angle in radians.
- Using the radius [tex]\(r = 3\)[/tex] meters and [tex]\(\theta \approx 1.2217\)[/tex] radians:
[tex]\[ L = 3 \times 1.2217 \approx 3.6652 \text{ meters} (approximated to 4 decimal places) \][/tex]
4. Round the Arc Length:
- Round the arc length to the nearest tenth of a meter:
[tex]\[ 3.6652 \approx 3.7 \text{ meters} \][/tex]
5. Conclusion:
- The approximate length of minor arc [tex]\(XZ\)[/tex] is [tex]\(3.7\)[/tex] meters.
So, the correct answer is:
[tex]\[ \boxed{3.7 \text{ meters}} \][/tex]
