To solve this problem, we need to follow these steps:
### Step 1: Calculate the Circumference of the Wheel
The circumference [tex]\(C\)[/tex] of a circle is given by the formula:
[tex]\[ C = 2 \pi r \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle and [tex]\(\pi\)[/tex] is a constant approximately equal to [tex]\(\frac{22}{7}\)[/tex].
Given:
- Radius [tex]\( r = 0.6 \, \text{m} \)[/tex]
- [tex]\(\pi = \frac{22}{7} \)[/tex]
Plug in the values:
[tex]\[ C = 2 \times \frac{22}{7} \times 0.6 \][/tex]
Perform the multiplication:
[tex]\[ C = 2 \times 1.8857142857142857 = 3.7714285714285714 \][/tex]
So, the circumference of the wheel is:
[tex]\[ C = 3.7714285714285714 \, \text{m} \][/tex]
### Step 2: Calculate the Angle in Radians
The angle in radians [tex]\(\theta\)[/tex] through which the wheel has turned can be found using the formula:
[tex]\[ \theta = \frac{\text{distance traveled}}{\text{circumference}} \][/tex]
Given the distance traveled along the ground is [tex]\(1.1 \, \text{m}\)[/tex], we have:
[tex]\[ \theta = \frac{1.1}{3.7714285714285714} \][/tex]
Perform the division:
[tex]\[ \theta = 0.2916666666666667 \][/tex]
So, the angle in radians is:
[tex]\[ \theta = 0.2916666666666667 \, \text{radians} \][/tex]
### Step 3: Convert the Angle from Radians to Degrees
To convert radians to degrees, we use the conversion factor [tex]\(\frac{180}{\pi}\)[/tex]:
Given:
[tex]\[ \theta_{\text{degrees}} = \theta_{\text{radians}} \times \left(\frac{180}{\pi}\right) \][/tex]
where [tex]\(\pi = \frac{22}{7}\)[/tex].
Plug in the known values:
[tex]\[ \theta_{\text{degrees}} = 0.2916666666666667 \times \left(\frac{180}{\frac{22}{7}}\right) \][/tex]
Simplify the conversion factor:
[tex]\[ \theta_{\text{degrees}} = 0.2916666666666667 \times \left(\frac{180 \times 7}{22}\right) \][/tex]
[tex]\[ \theta_{\text{degrees}} = 0.2916666666666667 \times 57.27272727272727 \][/tex]
Perform the multiplication:
[tex]\[ \theta_{\text{degrees}} = 16.704545454545457 \][/tex]
So, the angle in degrees is:
[tex]\[ \theta = 16.704545454545457^\circ \][/tex]
### Summary
- The circumference of the wheel is [tex]\(3.7714285714285714 \, \text{m}\)[/tex]
- The angle in radians through which the wheel has turned is [tex]\(0.2916666666666667\)[/tex]
- The angle in degrees through which the wheel has turned is [tex]\(16.704545454545457^\circ\)[/tex]
Thus, the wheel has turned through approximately [tex]\(16.7^\circ\)[/tex].
