Answer :
To solve the problem of finding the approximate length of the minor arc [tex]\( XZ \)[/tex] in circle [tex]\( Y \)[/tex] with a given central angle [tex]\( X Y Z \)[/tex], we need to follow these steps:
### Step 1: Understand the parameters
- Radius of the circle [tex]\( r \)[/tex]: [tex]\( 3 \)[/tex] meters.
- Central angle [tex]\( X Y Z \)[/tex]: [tex]\( 70^{\circ} \)[/tex].
### Step 2: Convert degrees to radians
The formula to convert degrees to radians is:
[tex]\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \][/tex]
Plugging in the values:
[tex]\[ 70^{\circ} \times \frac{\pi}{180} \approx 1.2217 \text{ radians} \][/tex]
### Step 3: Calculate the arc length
The formula for the arc length [tex]\( L \)[/tex] is:
[tex]\[ L = r \times \text{central angle (in radians)} \][/tex]
Using [tex]\( r = 3 \)[/tex] meters and the central angle in radians:
[tex]\[ L = 3 \times 1.2217 \approx 3.6652 \text{ meters} \][/tex]
### Step 4: Round the arc length to the nearest tenth of a meter
Rounding [tex]\( 3.6652 \)[/tex] meters to the nearest tenth:
[tex]\[ 3.7 \text{ meters} \][/tex]
### Conclusion
The approximate length of the minor arc [tex]\( XZ \)[/tex] is [tex]\( 3.7 \)[/tex] meters.
Thus, the correct answer is:
[tex]\[ \boxed{3.7 \text{ meters}} \][/tex]
### Step 1: Understand the parameters
- Radius of the circle [tex]\( r \)[/tex]: [tex]\( 3 \)[/tex] meters.
- Central angle [tex]\( X Y Z \)[/tex]: [tex]\( 70^{\circ} \)[/tex].
### Step 2: Convert degrees to radians
The formula to convert degrees to radians is:
[tex]\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \][/tex]
Plugging in the values:
[tex]\[ 70^{\circ} \times \frac{\pi}{180} \approx 1.2217 \text{ radians} \][/tex]
### Step 3: Calculate the arc length
The formula for the arc length [tex]\( L \)[/tex] is:
[tex]\[ L = r \times \text{central angle (in radians)} \][/tex]
Using [tex]\( r = 3 \)[/tex] meters and the central angle in radians:
[tex]\[ L = 3 \times 1.2217 \approx 3.6652 \text{ meters} \][/tex]
### Step 4: Round the arc length to the nearest tenth of a meter
Rounding [tex]\( 3.6652 \)[/tex] meters to the nearest tenth:
[tex]\[ 3.7 \text{ meters} \][/tex]
### Conclusion
The approximate length of the minor arc [tex]\( XZ \)[/tex] is [tex]\( 3.7 \)[/tex] meters.
Thus, the correct answer is:
[tex]\[ \boxed{3.7 \text{ meters}} \][/tex]
