Let's work through both parts of the problem one at a time:

Part (a)

You are asked to find the tire rotations per second when traveling at 60 miles per hour.

1. Convert speed to feet per second:

   $60 \, \text{miles/hour} = 60 \times 5280 \, \text{feet/mile} \div 3600 \, \text{seconds/hour} = 88 \, \text{feet/second}$

2. Find the total diameter of the wheel (metal + rubber):

   The total diameter is the metal part (13.5 inches) plus the rubber part (4.3 inches), so: $\text{Total diameter} = 13.5 + 4.3 = 17.8 \, \text{inches}$ The circumference of the tire is given by: $\text{Circumference} = \pi \times \text{diameter} = \pi \times 17.8 \approx 55.9 \, \text{inches}$

3. Convert the tire's circumference to feet:

   Since there are 12 inches in a foot: $55.9 \, \text{inches} \div 12 = 4.66 \, \text{feet}$

4. Calculate tire rotations per second:

   To find the rotations per second, divide the speed by the tire's circumference: $\text{Rotations per second} = \frac{88 \, \text{feet/second}}{4.66 \, \text{feet/rotation}} \approx 18.88 \, \text{rotations/second}$

Thus, the number of rotations per second is approximately 18.881.

Part (b)

Here, you change the tire size to 5.15 inches of rubber and need to determine how fast you were actually traveling if the speedometer was reading 65 mph.

1. Find the new total diameter:

   The new total diameter is the original metal part plus the new rubber size: $\text{New diameter} = 13.5 + 5.15 = 18.65 \, \text{inches}$

2. Find the new circumference:

   $\text{New circumference} = \pi \times 18.65 \approx 58.6 \, \text{inches}$

3. Convert this new circumference to feet:

   $58.6 \, \text{inches} \div 12 = 4.88 \, \text{feet}$

4. Calculate the actual speed:

   To find the actual speed, compare the ratio of the new circumference to the old one and multiply by the speedometer reading: $\text{Actual speed} = 65 \times \frac{4.88 \, \text{feet}}{4.66 \, \text{feet}} \approx 68.1 \, \text{mph}$

Thus, the actual speed is approximately 68 mph.

Would you like further clarification or details on any part of the solution?

Relative Questions:

1. How is the relationship between tire circumference and speed calculated?
2. Why is converting units (miles to feet, inches to feet) important in solving such problems?
3. How does a larger tire size affect speedometer readings?
4. What happens if you use tires smaller than the calibrated size?
5. How does tire rotation speed correlate with vehicle speed?

Tip:

Always ensure your speedometer is recalibrated after changing tire sizes to avoid miscalculations of your actual speed.