Imagine that there is a circular track given by the equation x2+y2=1123600, where both x and y are measured in units of feet. At time t=0, Robin is at point (1060,0) and Shannon is at point (560,900). Both are running counterclockwise, with Shannon running at 3.9 feet per second and Robin running at 6 feet per second.Robin runs until he catches Shannon. How long does that take? At what point do Robin and Shannon meet?
To solve this problem, we can find the time it takes for Robin to catch up to Shannon, as well as the coordinates of the point where they meet.
1. First, let's find the equations of motion for Robin and Shannon. Since Robin runs at a constant speed of 6 feet per second, we can write his position as a function of time, t, as follows:
Robin: x(t) = 1060 + 6t, y(t) = 0
Similarly, since Shannon runs at a constant speed of 3.9 feet per second, we have:
Shannon: x(t) = 560, y(t) = 900 - 3.9t
2. Now, we need to find the time when Robin catches up to Shannon. This occurs when their x-coordinates and y-coordinates are equal. Therefore, we set up the following equations:
1060 + 6t = 560
0 = 900 - 3.9t
Solving these equations, we find:
t = 250 seconds
Hence, it takes 250 seconds for Robin to catch up to Shannon.
3. Finally, to find the coordinates of the point where Robin and Shannon meet, we substitute the value of t into either Robin's or Shannon's equation. Let's use Robin's equation:
x(t) = 1060 + 6t
y(t) = 0
Substituting t = 250 seconds, we get:
x(250) = 1060 + 6(250) = 2560
y(250) = 0
Therefore, Robin and Shannon meet at the point (2560, 0) after 250 seconds.