MHB 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, $$s=r\theta$$
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
$$\theta$$ = angular displacement (and must be in radian form)
r = radius

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

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

 

[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, \theta[/b], in radians by multiplying 2\pi by 10000, and <b>then</b> multiply by r= 14 in.<br /> <br /> In other words, your method was to first multiply 2\pi r then multiply by 10000 while using &quot;r\theta&quot; you first find \theta by multiplying 10000(2\pi) and <b>then</b> 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 :)