Sure! Let's solve the problem step-by-step.
### Part i) Find the radius of the wheel
1. Given Information:
- The bicycle wheel makes 30 turns to cover a distance of 132 meters.
2. Calculate the circumference of the wheel:
The total distance covered by the wheel is equal to the number of turns multiplied by the circumference of the wheel. Thus, we can find the circumference (C) of one turn of the wheel as:
[tex]\[
C = \frac{\text{Total Distance}}{\text{Number of Turns}}
\][/tex]
Given the total distance is 132 meters and the number of turns is 30:
[tex]\[
C = \frac{132 \, \text{m}}{30} = 4.4 \, \text{m}
\][/tex]
3. Relate the circumference to the radius:
The circumference of a circle is also given by [tex]\(C = 2\pi r\)[/tex], where [tex]\(r\)[/tex] is the radius of the circle. Therefore, we can solve for the radius [tex]\(r\)[/tex]:
[tex]\[
r = \frac{C}{2\pi}
\][/tex]
Substituting the value of [tex]\(C\)[/tex]:
[tex]\[
r = \frac{4.4 \, \text{m}}{2\pi}
\][/tex]
Numerically, this evaluates to approximately:
[tex]\[
r \approx 0.7002817496043395 \, \text{m}
\][/tex]
### Part ii) Express the speed in km/h
1. Calculate the speed in meters per second:
Assuming the distance of 132 meters is covered in 1 second, the speed [tex]\(v\)[/tex] in meters per second (m/s) is:
[tex]\[
v = \frac{\text{Distance}}{\text{Time}}
\][/tex]
Given the distance is 132 meters and the time is 1 second:
[tex]\[
v = \frac{132 \, \text{m}}{1 \, \text{s}} = 132 \, \text{m/s}
\][/tex]
2. Convert the speed to kilometers per hour:
To convert from meters per second (m/s) to kilometers per hour (km/h), we use the conversion factor [tex]\(1 \, \text{m/s} = 3.6 \, \text{km/h}\)[/tex]:
[tex]\[
v \, \text{(km/h)} = v \, \text{(m/s)} \times 3.6
\][/tex]
Substituting the value of [tex]\(v\)[/tex]:
[tex]\[
v \, \text{(km/h)} = 132 \, \text{m/s} \times 3.6 = 475.2 \, \text{km/h}
\][/tex]
### Summary
- The radius of the wheel is approximately [tex]\(0.7003\)[/tex] meters.
- The speed of the bicycle in covering the distance is [tex]\(475.2 \, \text{km/h}\)[/tex].
