# Determining Miles Traveled From Tire Diameter and Rotations.

• MHB
• RidiculousName
In summary, the conversation was about finding the distance traveled by a car with tires of diameter 28" that rotate 10,000 times. The textbook provided a formula, s=r\theta, to find the relation between linear displacement and angular displacement. However, the question was about solving the problem using this formula. It was explained that for a complete rotation, \theta= 2\pi radians, and for a general angle of \theta there are \frac{\theta}{2\pi} rotations. Therefore, using the formula, the distance covered was calculated to be 280000\pi=879646 inches. However, it was also suggested to keep it simple by using the wheel circumference and the number of revolutions to find the
RidiculousName
I am trying to figure out how to solve this equation. I have a car with tires of diameter 28", and they rotate 10,000 times. How far did I travel?

According to my textbook it's 13.9 miles.

I can figure it out by finding the circumference of the tire (87.96"), multiplying that by 10,000 (879600), dividing the product by the amount of inches in a mile (63360) to get 13.8826.

But, I am supposed to do it with this formula, $$\displaystyle s=r\theta$$
However, I'm not sure how to do that at all.
It is a formula to find the relation between a linear displacement and an angular displacement.
s = linear displacement
$$\displaystyle \theta$$ = angular displacement (and must be in radian form)

I might be supposed to use $$\displaystyle v=r\omega$$
It is a formula to find the relation between a linear velocity and an angular velocity.
v = vertical speed
$$\displaystyle \omega$$ = angular speed (must be in radian form)

How can I solve this problem using the formula $$\displaystyle s=r\theta$$?

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For a complete rotation, $$\theta= 2\pi$$ radians. For a general angle of $$\theta$$ there are $$\frac{\theta}{2\pi}$$ rotations. Given a radius r, the circumference of a full circle is of course $$2\pi r$$ so that a rotation of $$\theta$$ radians gives a distance of $$2\pi r\frac{\theta}{2\pi}= r \theta$$.

Country Boy said:
For a complete rotation, $$\theta= 2\pi$$ radians. For a general angle of $$\theta$$ there are $$\frac{\theta}{2\pi}$$ rotations. Given a radius r, the circumference of a full circle is of course $$2\pi r$$ so that a rotation of $$\theta$$ radians gives a distance of $$2\pi r\frac{\theta}{2\pi}= r \theta$$.

Thank you, but I don't understand how that answers my question.

"10000 rotations" is $$\theta= 10000(2\pi)= 20000\pi$$ radians. The distance covered is $$\theta r= 20000\pi(14)= 280000\pi=879646$$ inches. That's just what you did, just with a slight change in the order of the multiplications. Your first calculated the circumfernce of the wheel, using $$2\pi r$$ then multiplied by 10000. Using $$r\theta$$ you first calculate the total angular rotation, $$\theta[/b], in radians by multiplying [tex]2\pi$$ by 10000, and then multiply by r= 14 in.

In other words, your method was to first multiply $$2\pi r$$ then multiply by 10000 while using "$$r\theta$$" you first find $$\theta$$ by multiplying $$10000(2\pi)$$ and then multiply by r= 14.

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What's wrong with keeping it simple:

u = wheel circumference = pi * 28 inches

10000 revolutions = u * 10000 inches

u * 10000 / (5280*12) = ~13.8833 miles

There's nothing wrong with that and, in fact, that was what the OP did. But his question was about using "$$r\theta$$" and that was what I was responding to.

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Yer right CB...should be more careful...
A thousand apologies of which you may have one :)

## 1. How do you determine the miles traveled from tire diameter and rotations?

To determine the miles traveled, you will need to know the diameter of the tire in inches and the number of rotations the tire has made. You can then use the following formula: miles = (diameter * rotations) / 63360. This will give you the distance traveled in miles.

## 2. What is the significance of tire diameter in determining miles traveled?

The tire diameter is an important factor in determining the miles traveled because it directly affects the distance the tire travels with each rotation. A larger tire diameter will cover more ground with each rotation, resulting in a greater distance traveled.

## 3. Can you use this method for all types of tires?

Yes, this method can be used for all types of tires as long as you have the diameter of the tire and the number of rotations. It is important to note that the formula may vary slightly for different types of tires, such as bicycle tires or tractor tires.

## 4. How accurate is this method for determining miles traveled?

This method is fairly accurate, but it may not account for factors such as tire wear or road conditions that can affect the distance traveled. It is always best to use this method as an estimate rather than an exact measurement.

## 5. Are there any other methods for determining miles traveled from tire diameter and rotations?

Yes, there are other methods that can be used to determine miles traveled, such as using GPS tracking devices or odometer readings. However, these methods may not be as accessible or convenient as using the tire diameter and rotations method.

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