Question
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 4 km/hour faster than the bicyclist heading south. At 10:15, they are 40 km apart. Find the two bicyclists’ rates.
northbound bicyclist = 16 km/h; southbound bicyclist = 10 km/h
northbound bicyclist = 19 km/h; southbound bicyclist = 15 km/h
northbound bicyclist = 18 km/h; southbound bicyclist = 13 km/h
northbound bicyclist = 18 km/h; southbound bicyclist = 14 km/h
northbound bicyclist = 16 km/h; southbound bicyclist = 10 km/h
northbound bicyclist = 19 km/h; southbound bicyclist = 15 km/h
northbound bicyclist = 18 km/h; southbound bicyclist = 13 km/h
northbound bicyclist = 18 km/h; southbound bicyclist = 14 km/h
Answers
GPT-4o mini
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:
1. Distance traveled by the northbound bicyclist = \( (r + 4) \times 1.25 \)
2. 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.**
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:
1. Distance traveled by the northbound bicyclist = \( (r + 4) \times 1.25 \)
2. 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.**