Let's denote the speed of the southbound bicyclist as \( x \) km/h. Since the northbound bicyclist is traveling 5 km/h faster, their speed will be \( x + 5 \) km/h.
The two bicyclists pass each other at 9:00 AM, and by 10:45 AM, 1 hour and 45 minutes (or 1.75 hours) later, they are 47.25 km apart.
The distance traveled by each bicyclist can be expressed in terms of time and speed.
For the southbound bicyclist:
- Distance = Speed × Time
- Distance = \( x \times 1.75 \) km
For the northbound bicyclist:
- Distance = Speed × Time
- Distance = \( (x + 5) \times 1.75 \) km
Since they are traveling in opposite directions, the total distance between them after 1.75 hours is the sum of the distances they have traveled:
\[ x \times 1.75 + (x + 5) \times 1.75 = 47.25 \]
Now let's simplify this equation:
\[ 1.75x + 1.75(x + 5) = 47.25 \] \[ 1.75x + 1.75x + 8.75 = 47.25 \] \[ 3.5x + 8.75 = 47.25 \]
Now, subtract 8.75 from both sides:
\[ 3.5x = 47.25 - 8.75 \] \[ 3.5x = 38.5 \]
Now, divide both sides by 3.5:
\[ x = \frac{38.5}{3.5} = 11 \]
So the speed of the southbound bicyclist is 11 km/h. Therefore, the northbound bicyclist's speed, being \( x + 5 \), is:
\[ 11 + 5 = 16 \text{ km/h} \]
Thus, the speeds of the bicyclists are:
- Northbound bicyclist: 16 km/h
- Southbound bicyclist: 11 km/h
The correct choice from your options is:
- northbound bicyclist = 16 km/h; southbound bicyclist = 11 km/h.