Question

6 (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?

Explanation:

Step1: Calculate the distance each cyclist travels

• For Maya:
• Speed \(v_{Maya}=6\space m/s\), time \(t = 2\times60 = 120\space s\)
• Distance \(s_{Maya}=v_{Maya}\times t=6\times120 = 720\space m\)
• For Jorge:
• Speed \(v_{Jorge}=4\space m/s\), time \(t = 2\times60 = 120\space s\)
• Distance \(s_{Jorge}=v_{Jorge}\times t=4\times120 = 480\space m\)

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

• Given \(r = 120\space m\)
• For Maya: \(\theta_{Maya}=\frac{s_{Maya}}{r}=\frac{720}{120}=6\space rad\)
• For Jorge: \(\theta_{Jorge}=\frac{s_{Jorge}}{r}=\frac{480}{120}=4\space rad\)

Answer:

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