To solve the problem of where vehicles A and B cross each other relative to the junction, we can break it down step by step. First, we need to establish their positions and velocities, then determine when and where they meet.
Setting Up the Problem
Let's define the positions of the vehicles:
- Vehicle A is 10 miles west of the junction, which we can denote as -10 miles.
- Vehicle B is 12 miles east of the junction, denoted as +12 miles.
Next, we note their velocities:
- Vehicle A travels towards the junction at 25 miles per hour.
- Vehicle B travels towards the junction at 40 miles per hour.
Calculating the Time Until They Meet
To find out when they will meet, we can set up an equation based on their distances from the junction and their speeds. We can express the distance each vehicle travels over time:
- Distance traveled by Vehicle A after time t: dA = 25t
- Distance traveled by Vehicle B after time t: dB = 40t
Since Vehicle A starts 10 miles away from the junction, its position relative to the junction after time t will be:
Position of A = -10 + 25t
For Vehicle B, starting 12 miles away, its position will be:
Position of B = 12 - 40t
Finding the Meeting Point
To find the time when they meet, we set their positions equal to each other:
-10 + 25t = 12 - 40t
Now, let's solve for t:
- Add 40t to both sides: 30t - 10 = 12
- Add 10 to both sides: 30t = 22
- Divide by 30: t = 22/30 = 11/15 hours
Calculating the Distance from the Junction
Now that we have the time, we can find out how far each vehicle has traveled towards the junction at that time:
- Distance traveled by Vehicle A: dA = 25 * (11/15) = 18.33 miles
- Distance traveled by Vehicle B: dB = 40 * (11/15) = 29.33 miles
Now, we need to determine the position of either vehicle at the time they meet. Let's use Vehicle A's position:
Position of A at t = 11/15 hours = -10 + 18.33 = 8.33 miles east of the junction.
Final Distance from the Junction
Thus, the distance from the junction to the point where they cross each other is:
8.33 miles east of the junction.
In summary, Vehicle A and Vehicle B will meet approximately 8.33 miles east of the junction. This approach illustrates how to set up equations based on motion and solve for the point of intersection effectively.
