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Let's break down the given problem:

1. **Tire Information:**
   - Wheel diameter = 15.5 inches (metal part).
   - Tire (rubber part) = 4.35 inches.

So, the **total tire diameter** is the sum of both parts:  
\[
\text{Total Diameter} = 15.5 \, \text{in} + 2(4.35 \, \text{in}) = 24.2 \, \text{inches}
\]
(The 4.35 inches is added twice, once for the top part of the tire and once for the bottom part).

2. **Part (a): Tire Rotations Per Second**
   - The car is moving at 40 miles per hour (mi/hr).
   - We need to find how many times the tire rotates per second.

### Step 1: Convert 40 miles per hour to inches per second.

1 mile = 63,360 inches.  
So, 40 miles per hour is:
\[
40 \, \text{mi/hr} = 40 \times 63,360 \, \text{in/hr} = 2,534,400 \, \text{in/hr}
\]
Now, convert hours to seconds:
\[
\frac{2,534,400 \, \text{in}}{3600 \, \text{seconds}} = 704 \, \text{in/sec}
\]

### Step 2: Calculate the circumference of the tire.
The formula for the circumference is:
\[
C = \pi \times \text{diameter}
\]
\[
C = 3.1416 \times 24.2 \, \text{in} = 76.045 \, \text{inches}
\]

### Step 3: Calculate the tire rotations per second.
Now, divide the distance traveled per second by the circumference of the tire:
\[
\text{Rotations per second} = \frac{704 \, \text{in/sec}}{76.045 \, \text{in}} = 9.258 \, \text{rotations/sec}
\]

Thus, the **rotations per second** is approximately **9.258**.

---

### Part (b): New Tires and Actual Speed

With the new tire size of 5.2 inches:
- New total diameter = 15.5 inches + 2(5.2 inches) = 25.9 inches.

Let's calculate the new speed.

### Step 1: Ratio of diameters
The speedometer is calibrated for a tire diameter of 24.2 inches, but you're using tires with a diameter of 25.9 inches. So, the ratio of the new to old diameters is:
\[
\frac{25.9}{24.2} = 1.070
\]

### Step 2: Calculate actual speed
The speedometer reads 60 mph, but the car is actually moving faster by the ratio we calculated:
\[
\text{Actual speed} = 60 \times 1.070 = 64.2 \, \text{mph}
\]
Rounded to the nearest whole number, the actual speed is **64 mph**.

---

Let me know if you'd like more details or have any questions!

Here are five related questions:
1. How would the calculation change if the diameter of the tire increased to 6 inches instead of 5.2?
2. What happens to the tire rotation frequency if the car’s speed doubles to 80 mph?
3. How does the tire circumference affect the overall fuel efficiency of a car?
4. Can you calculate the rotational speed in RPM (rotations per minute) for a speed of 50 mph?
5. What impact do under-inflated tires have on the speedometer's accuracy?

**Tip**: Always keep your tires inflated to the recommended pressure to ensure accurate speedometer readings and efficient fuel consumption.

Your car’s speedometer is geared to accurately give your speed using a certain tire size: 15.5" diameter wheels (the metal part) and 4.35" tires (the rubber part). (a) If your car’s instruments are properly calibrated, how many times should your tire rotate per second if you are travelling at 40 mi/hr? (b) You buy new 5.2" tires and drive at a constant speed of 60 mph. A cop claims you were speeding. How fast did the radar gun clock you?

Learn how to calculate tire rotations per second when driving at 40 mph with a specific tire size and determine the actual speed using different tires. This step-by-step guide covers concepts like tire circumference, speed conversion, and ratio of diameters, suitable for high school students.

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