To solve the problem, we need to calculate how many times each wheel rotates during the 14-mile journey and then find the difference between the number of rotations for the front and rear wheels.

Step 1: Convert miles to inches

First, we need to convert 14 miles into inches, since the wheel diameters are given in inches.

There are 63,360 inches in a mile. So, 14 miles is: $14 \times 63,360 = 887,040 \text{ inches}$

Step 2: Calculate the circumference of each wheel

The circumference of a circle (which represents one complete rotation of the wheel) is given by the formula: $C = \pi \times d$ where $d$ is the diameter of the circle.

• Front wheel diameter: 33 inches $C_{\text{front}} = \pi \times 33 \approx 103.672 \text{ inches}$

• Rear wheel diameter: 56 inches $C_{\text{rear}} = \pi \times 56 \approx 175.929 \text{ inches}$

Step 3: Calculate the number of rotations for each wheel

To find the number of rotations, divide the total distance traveled by the circumference of each wheel:

• Front wheel rotations: $\text{Rotations}_{\text{front}} = \frac{887,040}{103.672} \approx 8,554.39 \text{ rotations}$

• Rear wheel rotations: $\text{Rotations}_{\text{rear}} = \frac{887,040}{175.929} \approx 5,042.19 \text{ rotations}$

Step 4: Find the difference in rotations

Now, subtract the number of rear wheel rotations from the front wheel rotations to find the difference:

$\text{Difference} = 8,554.39 - 5,042.19 \approx 3,512.20$

Thus, the front wheel rotated approximately 3,512 more times than the rear wheel.

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Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How would the problem change if the wheel diameters were different?
2. What if the farmer drove 20 miles instead of 14 miles? How would the rotations change?
3. How do you calculate the number of rotations if the wheels are not perfectly circular?
4. What is the effect of increasing the wheel diameter on the number of rotations?
5. How does changing the terrain affect the number of wheel rotations?

Tip: When solving problems involving circles, always remember that the circumference is key to understanding rotation and distance relationships.