Determining Miles Traveled From Tire Diameter and Rotations.

[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)
r = radius

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)
r = radius

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

 

[HOI]

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 $$2\pi r\frac{\theta}{2\pi}= r \theta$$.

 

[RidiculousName]

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.

 

[HOI]

"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, [tex]\theta[/b], in radians by multiplying $$2\pi$$ 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.

 

[Wilmer]

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

 

[HOI]

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.

 

[Wilmer]

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