Question

1. (8 points) two cyclists, maya and jorge, ride around a circular truck 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.

Explanation:

Step1: Calculate the distance each cyclist travels

• For Maya:
• The formula for distance \(d = vt\), where \(v\) is speed and \(t\) is time.
• Given \(v_{Maya}=6\ m/s\) and \(t = 2\times60 = 120\ s\) (since \(2\) minutes \(= 2\times60\) seconds).
• Using the formula \(d_{Maya}=v_{Maya}\times t\), we have \(d_{Maya}=6\times120=720\ m\).
• For Jorge:
• Given \(v_{Jorge}=4\ m/s\) and \(t = 120\ s\).
• Using the formula \(d_{Jorge}=v_{Jorge}\times t\), we have \(d_{Jorge}=4\times120 = 480\ m\).

Step2: Use the formula for arc - length \(s = r\theta\) (where \(s\) is arc - length, \(r\) is radius, and \(\theta\) is the central angle in radians)

• For Maya, from \(s = r\theta\), we can solve for \(\theta\). Given \(r = 120\ m\) and \(s=d_{Maya}=720\ m\).
• \(\theta_{Maya}=\frac{d_{Maya}}{r}\), substituting the values \(\theta_{Maya}=\frac{720}{120}=6\) radians.
• For Jorge, given \(r = 120\ m\) and \(s = d_{Jorge}=480\ m\).
• \(\theta_{Jorge}=\frac{d_{Jorge}}{r}\), substituting the values \(\theta_{Jorge}=\frac{480}{120}=4\) radians.

Answer:

Maya has traveled \(6\) radians and Jorge has traveled \(4\) radians.