To solve the problem of when and where vehicles A and B will meet, we need to analyze their positions and speeds relative to the junction. Let's break it down step by step.
Understanding the Positions
Vehicle A starts 10 miles west of the junction, while vehicle B starts 12 miles east of it. We can visualize this scenario on a number line where the junction is at 0 miles:
- Vehicle A: -10 miles (west)
- Vehicle B: +12 miles (east)
Defining the Motion
Both vehicles are moving towards the junction:
- Vehicle A travels at 25 miles per hour (mi/h).
- Vehicle B travels at 40 miles per hour (mi/h).
Setting Up the Equations
We can express the positions of both vehicles as functions of time (t), where t is measured in hours:
- Position of Vehicle A: P_A(t) = -10 + 25t
- Position of Vehicle B: P_B(t) = 12 - 40t
Here, the negative sign for Vehicle A indicates it is moving towards the junction (increasing its position towards 0), while the positive sign for Vehicle B indicates it is also moving towards the junction (decreasing its position towards 0).
Finding the Meeting Point
To find the time when both vehicles meet, we set their position equations 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 the distance from the junction to the point where they cross. We can substitute t back into either position equation. Let's use Vehicle A's equation:
P_A(11/15) = -10 + 25(11/15)
Calculating this gives:
- 25(11/15) = 275/15 = 18.33 miles
- P_A(11/15) = -10 + 18.33 = 8.33 miles
Final Result
Thus, the distance from the junction to the point where the vehicles cross each other is approximately 8.33 miles east of the junction. This means that they will meet before reaching the junction, with Vehicle A having traveled a distance of about 18.33 miles towards it, and Vehicle B having traveled about 8 miles towards it.
