To find the rates of the two bicyclists, we need to set up some variables and equations based on the information provided.
Let \( r \) represent the speed of the southbound bicyclist (in km/h). Then, the speed of the northbound bicyclist, who is riding 4 km/h faster, can be represented as \( r + 4 \).
The two bicyclists pass each other at 9:00 AM and are 40 km apart at 10:15 AM, which is 1 hour and 15 minutes, or \( 1.25 \) hours later.
The distance traveled by both bicyclists can be expressed as:
- Distance traveled by the northbound bicyclist = \( (r + 4) \times 1.25 \)
- Distance traveled by the southbound bicyclist = \( r \times 1.25 \)
Since they are traveling in opposite directions, their total distance apart at 10:15 AM is the sum of the distances each traveled:
\[ (r + 4) \times 1.25 + r \times 1.25 = 40 \]
We can factor out \( 1.25 \) from the equation:
\[ 1.25 (r + 4 + r) = 40 \]
This simplifies to:
\[ 1.25 (2r + 4) = 40 \]
Dividing both sides by \( 1.25 \):
\[ 2r + 4 = \frac{40}{1.25} \]
Calculating \( \frac{40}{1.25} \):
\[ \frac{40}{1.25} = 32 \]
Now we have:
\[ 2r + 4 = 32 \]
Subtracting 4 from both sides:
\[ 2r = 28 \]
Dividing by 2:
\[ r = 14 \]
Now that we have the rate of the southbound bicyclist (14 km/h), we can find the rate of the northbound bicyclist:
\[ r + 4 = 14 + 4 = 18 \text{ km/h} \]
So, the rates of the two bicyclists are:
- Northbound bicyclist: 18 km/h
- Southbound bicyclist: 14 km/h
Thus, the correct answer is: northbound bicyclist = 18 km/h; southbound bicyclist = 14 km/h.