Final answer:
The runner's speed is 4 mph, and the cyclist's speed is 11 mph, derived by setting up equations from the information that the cyclist is 63 miles ahead of the runner after 9 hours and that the cyclist's speed is three mph more than twice the runner's speed.
Step-by-step explanation:
To find the speed of the runner and the cyclist, we'll use the given information to set up equations. We know the cyclist is 63 miles ahead after 9 hours and that the cyclist's speed is three mph more than twice the speed of the runner.
Let's denote the runner's speed as r miles per hour. Then, the cyclist's speed is 2r + 3 miles per hour. After 9 hours, the distances they traveled are 9r for the runner and 9(2r + 3) for the cyclist. Since the cyclist is 63 miles ahead:
9(2r + 3) - 9r = 63
By simplifying:
18r + 27 - 9r = 63
9r + 27 = 63
Subtracting 27 from both sides:
9r = 36
Dividing by 9:
r = 4 mph (runner's speed)
Now we can find the cyclist's speed:
2r + 3 = 2(4) + 3 = 11 mph (cyclist's speed)
Thus, the runner's speed is 4 mph and the cyclist's speed is 11 mph.