# Thread: People Moving Around Circle Track

1. ## People Moving Around Circle Track

Bob and Sally are each going for a run around a circlar track of radius 80 feet. They start running from different locations on the circle.
Bob starts at the easternmost point and jogs 10 feet/second around the track in the clockwise direction.
Sally runs counterclockwise at 14 feet/second and passes Bob for the first time in 16 seconds.
(Words I bolded are the given information we need)

Find Sally's x and y coordinates in 12 minutes.
So here is what I'm thinking on a coordinate plan
Bob starts at the easternmost point (meaning all the way on the right), at (80,0)
Sally is somewhere
Circumference: 2πr = 2π(80) = 160π feet, π is pi by the way. This is the full length track.
Though, I'm still confused on how to start out.
Can someone help?
Thanks

2. Can you write an expresion to locate Bob an any moment?

Hint: < 80*cos(t) , 80*sin(t) >

3. bob and Sally are each going for a run around a circlar track of radius 80 feet. They start running from different locations on the circle.
Bob starts at the easternmost point and jogs 10 feet/second around the track in the clockwise direction.
Sally runs counterclockwise at 14 feet/second and passes Bob for the first time in 16 seconds.
(Words I bolded are the given information we need)

Find Sally's x and y coordinates in 12 minutes.
So here is what I'm thinking on a coordinate plan
Bob starts at the easternmost point (meaning all the way on the right), at (80,0)
Sally is somewhere
Circumference: 2πr = 2π(80) = 160π feet, π is pi by the way. This is the full length track.
Though, I'm still confused on how to start out.
Can someone help?

In the first 16 seconds, Bob travels 10(16) = 160ft. around the track, clockwise, to a point B = 114.592º from the eastern most point on the track, A = (80,0)

In the same 16 seconds, Sally travels 14(16) = 224ft. from her starting point, C = (224+160) = 384ft., 275.022º counterclockwise from Bob's starting point.

Having Sally's starting position, it remains only to find how many circuits of the track she makes in 12 minutes from t = 0 to point D, 12(60)14/2(80)3.14 = 20.0537 revolutions from t = 0.

4. Originally Posted by TchrWill
bob and Sally are each going for a run around a circlar track of radius 80 feet. They start running from different locations on the circle.
Bob starts at the easternmost point and jogs 10 feet/second around the track in the clockwise direction.
Sally runs counterclockwise at 14 feet/second and passes Bob for the first time in 16 seconds.
(Words I bolded are the given information we need)

Find Sally's x and y coordinates in 12 minutes.
So here is what I'm thinking on a coordinate plan
Bob starts at the easternmost point (meaning all the way on the right), at (80,0)
Sally is somewhere
Circumference: 2πr = 2π(80) = 160π feet, π is pi by the way. This is the full length track.
Though, I'm still confused on how to start out.
Can someone help?

In the first 16 seconds, Bob travels 10(16) = 160ft. around the track, clockwise, to a point B = 114.592º from the eastern most point on the track, A = (80,0)

In the same 16 seconds, Sally travels 14(16) = 224ft. from her starting point, C = (224+160) = 384ft., 275.022º counterclockwise from Bob's starting point.

Having Sally's starting position, it remains only to find how many circuits of the track she makes in 12 minutes from t = 0 to point D, 12(60)14/2(80)3.14 = 20.0537 revolutions from t = 0.
Just wondering, how did you get the degrees

5. Originally Posted by Chaim
Just wondering, how did you get the degrees
Working mathematicians as a rule do any longer use the concept of degree.
You see that a radian is a clearly defined number over-against the concept of a degree.

6. Just wondering, how did you get the degrees

The circumference of the track is 2(80)3.14 = 502.65ft.

A to B is 160ft.
160/502.65 = .3183 x 360º = 114.59º
A to C
(224+160) = 384ft.
(384/502.65)360º = 275.022º counterclockwise from Bob's starting point.

7. Oh!
I see now, ok thanks!

8. Originally Posted by JeffM
In what sense is $2\pi$ more clearly defined than $360.$
clarity does not admit of degree; a definition is either clear or ambiguous. In practical terms, all measurements are approximate except by accident. Platonically, any division of a circle by a distinct real number seems to me to be equally exact and equally well defined. What privileges 2 * pi over 360 as a real number? If some real numbers are more clearly defined than others, what are the criteria that measure degrees of "clear definition" for real numbers? I am currently writing something on the appropriate rhetoric for dealing with numeric approximations in historical writing, which is difficult enough on the assumption that numbers as Platonic ideals are clearly defined, but the whole exercise makes no sense at all if some numbers are not clearly defined in even a Platonic sense.
In my reply I made no mention of either 360 nor $2\pi$, let alone anything about their ontological status.

The idea of dividing one revelation into 360 parts (we now call degrees) surely dates from a time the Babylonians of 2000-1600 BCE. As their astronomy matured and good records were kept, they found one complete revolution could be divided in 24 equal parts hence hours and 360 came out of the way they measured distance. So the 360 is an accident of history.

On the other hand, a radian has a very precise definition. We know a great deal about measurement of arc length on a circle. We have long known that the whole circle has arc length $d\cdot\pi$ where $d$ is the distance across the circle through the center. $\tfrac{d}{2}=r$ is radius. So a radian is the measure of the central angle in circle which subtends an arc of length equal to the radius. Thus the length of the whole circle is $2\cdot\pi\cdot r$ radians. There is nothing arbitrary (no accident) there.

As a sidebar, much has been made of the fact that the value of $\pi$ shows up in measurements in old Egypt, say the pyramids. Of course the mystery goes away when one realizes that ancient Egyptians used a wheel with a peg in it to measure distance.

9. Originally Posted by pka
In my reply I made no mention of either 360 nor $2\pi$, let alone anything about their ontological status.

The idea of dividing one revelation into 360 parts (we now call degrees) surely dates from a time the Babylonians of 2000-1600 BCE. As their astronomy matured and good records were kept, they found one complete revolution could be divided in 24 equal parts hence hours and 360 came out of the way they measured distance. So the 360 is an accident of history.

On the other hand, a radian has a very precise definition. We know a great deal about measurement of arc length on a circle. We have long known that the whole circle has arc length $d\cdot\pi$ where $d$ is the distance across the circle through the center. $\tfrac{d}{2}=r$ is radius. So a radian is the measure of the central angle in circle which subtends an arc of length equal to the radius. Thus the length of the whole circle is $2\cdot\pi\cdot r$ radians. There is nothing arbitrary (no accident) there.

As a sidebar, much has been made of the fact that the value of $\pi$ shows up in measurements in old Egypt, say the pyramids. Of course the mystery goes away when one realizes that ancient Egyptians used a wheel with a peg in it to measure distance.
"An accident of history" has nothing to do with precision. The "degree" is as precisely defined as "radian".

Yes, when we are working with Calculus formulas, radians are preferable because the formulas for derivatives of the trig functions are much simpler in terms of radians (if x is in radians, (sin(x))'= cos(x) if x is in degrees, $(sin(x))'= \frac{\pi}{180} cos(x)$) but in problems that do not require derivatives or integrals, such as this one, one is as good as the other.

10. Originally Posted by HallsofIvy
"An accident of history" has nothing to do with precision. The "degree" is as precisely defined as "radian".
Once again, I did not say that that. I said the choice of 360 was an accident of history, at least that is the theory put forth by Otto Neugebauer.

BTW. Did you read the question in reply #8: "In what sense is
2πmoreclearlydefinedthan360?".

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