Question

1. (8 points) two cyclists, maya and jorge, ride around a circular track with a radius of 120 m. they both start at the same point at the same time.

• maya rides clockwise at a constant speed of 6 m/s.
• jorge rides counterclockwise at a constant speed of 4 m/s.

after 2 minutes, the coach wants to know:
(a) how many radians has each cyclist traveled around the circle?
(b) what is the central angle θ (in radians) between them at that instant?
(c) find the shorter arc length of the track separating maya and jorge.
(d) if the tracks total length is 2πr, what fraction of the track lies between them along that shorter arc?

Explanation:

Step1: Calculate the distance each cyclist travels

First, convert 2 minutes to seconds. Since \(1\) minute \( = 60\) seconds, \(t=2\times60 = 120\) seconds.
For Maya: Using the formula \(d = vt\) (where \(v\) is speed and \(t\) is time), \(d_{Maya}=6\times120 = 720\)m.
For Jorge: \(d_{Jorge}=4\times120 = 480\)m.

Step2: Use the formula \(\theta=\frac{s}{r}\) to find the angle in radians

The formula for the central - angle \(\theta\) (in radians) is \(\theta=\frac{s}{r}\), where \(s\) is the arc - length and \(r\) is the radius of the circle. Given \(r = 120\)m.
For Maya: \(\theta_{Maya}=\frac{720}{120}=6\) radians.
For Jorge: \(\theta_{Jorge}=\frac{480}{120}=4\) radians.

Step3: Calculate the central angle \(\theta\) between them

Since they are moving in opposite directions, \(\theta=\theta_{Maya}+\theta_{Jorge}\bmod{2\pi}\). \(\theta_{Maya}+\theta_{Jorge}=6 + 4=10\) radians. \(10-2\pi\approx10 - 6.28 = 3.72\) radians (because \(2\pi\approx6.28\) and we want the angle in the range \([0,2\pi]\)).

Step4: Calculate the shorter arc length

Using the formula \(s = r\theta\), with \(r = 120\)m and \(\theta\approx3.72\) radians. \(s=120\times3.72 = 446.4\)m.

Step5: Calculate the fraction of the track

The total length of the track \(C = 2\pi r=2\pi\times120\). The fraction \(f=\frac{s}{C}=\frac{3.72}{2\pi}\approx\frac{3.72}{6.28}\approx0.592\).

Answer:

(a) Maya: \(6\) radians, Jorge: \(4\) radians.
(b) Approximately \(3.72\) radians.
(c) Approximately \(446.4\)m.
(d) Approximately \(0.592\).