Let's first find out how far the cyclist has traveled in 3 hours at a speed of 12 mph:
distance = speed x time
distance = 12 mph x 3 hours
distance = 36 miles
So, the cyclist has traveled 36 miles in 3 hours.
Now, let's find out how long it takes for the car to catch up to the cyclist:
Let t be the time it takes for the car to catch up to the cyclist.
distance traveled by car = distance traveled by cyclist + 36 miles (the head start distance).
Using the formula distance = speed x time, we can express the distance traveled by the car and the cyclist as follows:
distance traveled by car = 48 mph x t
distance traveled by cyclist = 12 mph x t
We can now set up the equation:
48t = 12t + 36
Simplifying the equation:
48t - 12t = 36
36t = 36
t = 1 hour
Therefore, it takes 1 hour for the car to catch up to the cyclist.
To find the distance from the starting point where the car overtakes the cyclist, we can substitute t = 1 hour in the formula:
distance traveled by car = 48 mph x 1 hour = 48 miles
So, the car overtakes the cyclist at a distance of 48 miles from the starting point.