At 9:00 on Saturday morning, two bicyclists heading in opposite directions pass each other on a bicycle path. The bicyclist heading north is riding 5 km/hour faster than the bicyclist heading south. At 10:30, they are 40.5 km apart. find both of their rates.
To find the rates of both bicyclists, we need to set up a system of equations based on the given information.
Let's assume that the rate of the bicyclist heading south is r km/hour. Since the bicyclist heading north is riding 5 km/hour faster, their rate will be r + 5 km/hour.
We can use the formula distance = rate × time to represent the distance traveled by each bicyclist.
The bicyclist heading south traveled for 1.5 hours (from 9:00 to 10:30) and the distance is r × 1.5 km.
The bicyclist heading north also traveled for 1.5 hours, but at a faster rate of (r + 5) km/hour, so the distance is (r + 5) × 1.5 km.
According to the problem, the sum of these distances is 40.5 km:
r × 1.5 + (r + 5) × 1.5 = 40.5
Now, we can solve this equation to find the value of r, which represents the rate of the bicyclist heading south:
1.5r + 1.5(r + 5) = 40.5
1.5r + 1.5r + 7.5 = 40.5
3r + 7.5 = 40.5
3r = 33
r = 11
Therefore, the rate of the bicyclist heading south is 11 km/hour. Since the rate of the bicyclist heading north is r + 5, their rate would be 11 + 5 = 16 km/hour.
Since they are heading north and south, let the southern speed be s, so the northern speed is s+5
After 1.5 hours, the distance is 40.5km
distance is speed * time
40.5 = (s + s+5) * 1.5
27 = 2s+5
22 = 2s
s = 11
so, the cyclist going south moves at 11km/hr, north at 16 km/hr
they are moving apart at 27km/hr, so at 1.5 hours they are 40.5 km apart.