Sure, let's solve this step by step.
### Part (a) - Calculating the Circumference
The formula to calculate the circumference \(C\) of a circle is given by:
[tex]\[ C = 2 \pi r \][/tex]
where:
- \( \pi \approx 3.14159 \)
- \( r \) is the radius of the circle
In this problem, the radius \(r\) of the wheel is 35 cm.
Plugging in the values:
[tex]\[ C = 2 \times \pi \times 35 \][/tex]
This gives us the circumference:
[tex]\[ C \approx 219.9114857512855 \, \text{cm} \][/tex]
So, the circumference of the wheel is approximately \(219.91\) cm.
### Part (b) - Calculating the Total Distance Traveled
To find out how far the car travels after the wheel has rotated a certain number of times, we must multiply the circumference of the wheel by the number of rotations.
Given:
- The wheel rotates 100,000 times
- The circumference of the wheel \(C \approx 219.9114857512855 \, \text{cm}\)
Total distance traveled [tex]\[ D \][/tex] is calculated as follows:
[tex]\[ D = \text{Circumference} \times \text{Number of Rotations} \][/tex]
Plugging in the values:
[tex]\[ D = 219.9114857512855 \, \text{cm} \times 100000 \][/tex]
This gives us:
[tex]\[ D \approx 21991148.57512855 \, \text{cm} \][/tex]
Therefore, the car travels approximately \(21991148.57512855\) cm after the wheel rotates 100,000 times.
Converting this distance to meters, since 1 meter = 100 cm, we have:
[tex]\[ D \approx 219911.4857512855 \, \text{meters} \][/tex]
So, the car travels approximately \(219911.49\) meters or around \(219.91\) kilometers.
### Summary
(a) The circumference of the wheel is approximately \(219.91\) cm.
(b) After 100,000 rotations, the car travels approximately [tex]\(21991148.57512855\)[/tex] cm, which is about [tex]\(219.91\)[/tex] kilometers.
