Imagine that there is a circular track given by the equation x2+y2=2102500, where both x and y are measured in units of feet. At time t=0, Robin is at point (1450,0) and Shannon is at point (−170,1440). Both are running counterclockwise, with Shannon running at 2.3 feet per second and Robin running at 3.6 feet per second.

Question

Imagine that there is a circular track given by the equation x2+y2=2102500, where both x and y are measured in units of feet. At time t=0, Robin is at point (1450,0) and Shannon is at point (−170,1440). Both are running counterclockwise, with Shannon running at 2.3 feet per second and Robin running at 3.6 feet per second.

Robin runs until he catches Shannon. How long does that take?

At what point do Robin and Shannon meet?

Answer 1

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Answer 2

To find the time it takes for Robin to catch Shannon, we need to determine their relative speeds. Since they are both moving counterclockwise, their relative speed is the difference between their individual speeds.

Relative speed = Robin's speed - Shannon's speed
Relative speed = 3.6 ft/s - 2.3 ft/s
Relative speed = 1.3 ft/s

Given that they start at points (1450,0) and (-170,1440) respectively, we can see that they start at different positions on the circular track with a radius of √2102500 ft.

Next, we need to find the angle at which they start in order to determine their starting positions on the circle. We can use trigonometry to find this angle.

Angle = arctan(y / x) where (x, y) is the starting position of each runner

Angle of Robin = arctan(0 / 1450) = 0 degrees
Angle of Shannon = arctan(1440 / -170) = 262.91 degrees

Now, we can find the difference in angles between Robin and Shannon, which represents the head start that Shannon has.

Angle difference = Angle of Shannon - Angle of Robin
Angle difference = 262.91 degrees - 0 degrees
Angle difference = 262.91 degrees

To determine the time it takes for Robin to catch Shannon, we need to find how long it takes for Robin to cover the head start distance on the circular track.

Distance covered by Robin = 2πR * (Angle difference / 360 degrees)
Distance covered by Robin = 2π * √2102500 ft * (262.91 degrees / 360 degrees)

After calculating the above expression, we will get the distance covered by Robin.

Finally, we can find the time it takes for Robin to catch Shannon by dividing the distance covered by Robin by the relative speed.

Time taken = Distance covered by Robin / Relative speed

Substituting the values into the equation, we can find the time it takes for Robin to catch Shannon.