To solve this problem, we'll follow a few key steps to find the angle through which the wheel has turned as it rolls.
### Step 1: Calculate the Circumference of the Wheel
First, we need to determine the circumference of the wheel, which is the distance the wheel would travel if it made one complete revolution.
The formula for the circumference [tex]\( C \)[/tex] of a circle is given by:
[tex]\[ C = 2 \pi r \][/tex]
Given:
- Radius [tex]\( r = 0.6 \)[/tex] meters
- [tex]\(\pi = \frac{22}{7}\)[/tex]
So,
[tex]\[ C = 2 \times \frac{22}{7} \times 0.6 \][/tex]
[tex]\[ C = 2 \times 22 \times 0.6 \div 7 \][/tex]
[tex]\[ C \approx 3.7714285714285714 \text{ meters} \][/tex]
### Step 2: Calculate the Angle Turned in Radians
Next, we determine the angle the wheel has turned through in radians as it rolls the given distance of 1.1 meters.
Using the formula for the arc length, which is:
[tex]\[ \text{Arc Length} = r \times \text{Angle in Radians} \][/tex]
We can rearrange this to find the angle in radians:
[tex]\[ \text{Angle in Radians} = \frac{\text{Arc Length}}{r} \][/tex]
Plugging in the given values:
- Distance (Arc Length) = 1.1 meters
- Radius [tex]\( r = 0.6 \)[/tex] meters
So,
[tex]\[ \text{Angle in Radians} = \frac{1.1}{0.6} \][/tex]
[tex]\[ \text{Angle in Radians} \approx 1.8333333333333335 \][/tex]
### Step 3: Convert the Angle from Radians to Degrees
Lastly, we convert the angle from radians to degrees. The conversion formula is:
[tex]\[ \text{Angle in Degrees} = \text{Angle in Radians} \times \left(\frac{180}{\pi}\right) \][/tex]
Given:
- [tex]\(\pi = \frac{22}{7}\)[/tex]
- Angle in Radians [tex]\(\approx 1.8333333333333335\)[/tex]
So,
[tex]\[ \text{Angle in Degrees} = 1.8333333333333335 \times \left(\frac{180}{\frac{22}{7}}\right) \][/tex]
[tex]\[ \text{Angle in Degrees} \approx 1.8333333333333335 \times 57.27272727272727 \][/tex]
[tex]\[ \text{Angle in Degrees} \approx 105.04226244065093 \][/tex]
### Conclusion
The result from these calculations is:
- The circumference of the wheel is approximately [tex]\( 3.7714285714285714 \)[/tex] meters.
- The angle turned by the wheel in radians is approximately [tex]\( 1.8333333333333335 \)[/tex].
- The angle turned by the wheel in degrees is approximately [tex]\( 105.04226244065093 \)[/tex].
Thus, the wheel has turned through an angle of approximately [tex]\( 105.04226244065093 \)[/tex] degrees as it rolled a distance of 1.1 meters.
