Towns A and B are 420km apart. Two lorries departed from A at the same time travelling towards B. Lorry x travelled at an average speed of 15km/h less than Y and reached 1 hour 24 minutes later.
c) A van left town B heading towards A at the time lorries x and y left A. If the van travelled at an average speed of 90km/h, how far from A did it meet lorry Y
To find the distance from A where the van met lorry Y, we first need to determine the time it took for lorry Y to reach that point.
Let the average speed of lorry Y be represented by V.
The speed of lorry X is V - 15 km/h.
Let the time it took for lorry Y to reach the meeting point be represented by T hours.
Since distance = speed * time, we have the following equations:
420 = V * T (1)
420 = (V - 15) * (T + 1 2/3) (2)
Let's solve equation (2) to find the values of V and T.
420 = (V - 15) * (1.4T + 1 2/3) [Converting 1 hour 24 minutes to 1.4 hours]
420 = (V - 15) * (7T + 10)/5 [Converting fractions to a common denominator of 5]
5 * 420 = (V - 15) * (7T + 10) [Multiplying both sides by 5]
2100 = (V - 15) * (7T + 10)
Since this equation has two unknowns, V and T, it cannot be solved directly. However, we are given additional information about the van.
The van traveled at an average speed of 90 km/h. Let the time it took for the van to meet lorry Y be represented by T_van hours. The distance traveled by the van is 90 * T_van = D.
From A, the lorries travel a distance of 420 - D before meeting each other. Using equation (1), we can write:
420 - D = V * (T - T_van)
From B, the van travels a distance of 420 - D before meeting lorry Y. Using equation (1), we can write:
420 - D = 90 * (T - T_van)
Setting the two expressions for 420 - D equal to each other, we have:
V * (T - T_van) = 90 * (T - T_van)
Simplifying further:
VT - VT_van = 90T - 90T_van
VT + 90T_van = 90T + VT_van
V(T - T_van) = 90(T - T_van)
Since T - T_van ≠ 0 (because they are different times), we can divide both sides of the equation by (T - T_van):
V = 90
This means that the average speed of lorry Y is 90 km/h.
Now let's go back to equation (1) and substitute V = 90:
420 = 90T
T = 420/90 = 14/3 = 4 2/3 hours
So lorry Y took 4 hours and 40 minutes to reach the meeting point.
Now, let's find the distance from A where lorry Y and the van met. The van traveled at an average speed of 90 km/h for 4 2/3 hours:
D = 90 * (14/3) = 420 km
Therefore, the van met lorry Y 420 km from A.
To find the distance at which the van met lorry Y, we need to find the time it took for lorry Y to travel from town B to the meeting point.
Let's assume that lorry Y traveled from town A to the meeting point in t hours. Since the distance between towns A and B is 420 km, lorry Y traveled t hours at a speed of (90 + 15) km/h = 105 km/h.
Therefore, the distance traveled by lorry Y is:
Distance = Speed * Time
Distance = 105 km/h * t
Now, we know that lorry X arrived 1 hour and 24 minutes later than lorry Y at the meeting point. We need to convert 1 hour and 24 minutes to hours. There are 60 minutes in 1 hour, so:
1 hour and 24 minutes = 1 + 24/60 = 1 + 0.4 = 1.4 hours
Since lorry X travels at an average speed of 15 km/h less than lorry Y, its average speed is (105 - 15) km/h = 90 km/h.
Using lorry X's average speed, we can calculate the time it took for lorry X to travel from town A to the meeting point:
Time = Distance / Speed
Time = 90 km/h * (t + 1.4 hours)
We know that the total distance between towns A and B is 420 km, so:
Distance_Y + Distance_X = 420 km
Substituting the distances with their respective equations:
105t + 90(t + 1.4) = 420
Simplifying the equation:
105t + 90t + 126 = 420
195t + 126 = 420
195t = 420 - 126
195t = 294
t = 294 / 195
t = 1.51 hours
Therefore, lorry Y took approximately 1.51 hours to travel from town B to the meeting point.
Now, let's find the distance from town A to the meeting point:
Distance_Y = Speed * Time
Distance_Y = 105 km/h * 1.51 hours
Distance_Y ≈ 158.55 km
Hence, lorry Y met the van approximately 158.55 km from town A.