Two towns P and Q are 400 km apart. A bus left for P and Q. It stopped at Q for one hour and then started the return to P. one hour after the departure of the bus from P, a trail also heading for Q left P. the trailer met the returning bus ¾ of the way from P to Q. they met t hours after the departure of the bus from P.
(a) Express the average speed of the trailer in terms of t
(b) Find the ratio of the speed of the bus so that of the trailer.
The trailer went 300 km in t-1 hours, so its avg speed is 300/(t-1)
Let the bus's speed be x, and the trailer's speed be y.
The bus went 500 km.
Its time spent on the trip before meeting the trailer was t = 400/x + 1 + 100/x = 1 + 500/x hours
The trailer was on the road for 300/y hours
so, 300/y = 500/x
(b) x/y = 5/3
check:
x=50, y=30
400/50 + 1 + 100/50 = 11 hrs
300/30 = 10, or 1 hours less
To solve this problem, we need to analyze the distance traveled by both the bus and the trailer, as well as the time taken by each of them. Let's break it down step by step:
1. The distance between towns P and Q is 400 km.
2. The bus leaves for town P and then continues to town Q. Let's call the speed of the bus "v_bus".
3. The bus stops at town Q for one hour. This stoppage does not affect the average speed of the bus for the entire trip.
4. After the one-hour stop at Q, the bus starts its return journey from Q to P.
5. One hour after the bus departs from P, the trailer leaves P for town Q. Let's call the speed of the trailer "v_trailer".
6. The trailer meets the returning bus ¾ of the way from P to Q. This means that the distance traveled by the bus at that point is ¾ * 400 km = 300 km.
7. They meet "t" hours after the departure of the bus from P.
Now, let's calculate the average speed of the trailer in terms of "t" based on the information given:
Distance traveled by the trailer = Distance between P and Q - Distance traveled by the bus = 400 km - 300 km = 100 km.
Since the trailer meets the returning bus "t" hours after the departure of the bus from P, we can express the time taken by the trailer as "t".
Average speed of the trailer = Distance traveled by the trailer ÷ Time taken = 100 km ÷ t hours = 100/t km/h.
So, the average speed of the trailer in terms of "t" is 100/t km/h.
To find the ratio of the speed of the bus to that of the trailer, we need to know the speed of the bus in relation to that of the trailer.
Let's call the ratio of the speed of the bus to that of the trailer "r".
Speed of the bus = r * Speed of the trailer = r * v_trailer.
Based on the information given, we know that the distance traveled by both the bus and the trailer is the same, i.e., 400 km. Therefore, the time taken by the bus should be smaller than that taken by the trailer.
Time taken by the bus = 400 km ÷ v_bus.
Since the trailer meets the bus "t" hours after its departure, the time taken by the trailer is "t" hours more than that taken by the bus.
Time taken by the trailer = Time taken by the bus + t = 400 km ÷ v_bus + t.
Now, let's set up an equation based on the information above:
Distance traveled by the bus = Distance traveled by the trailer.
Speed of the bus * Time taken by the bus = Speed of the trailer * Time taken by the trailer.
v_bus * (400 km ÷ v_bus) = v_trailer * (400 km ÷ v_bus + t).
Simplifying the equation:
400 km = v_trailer * (400 km ÷ v_bus + t).
Divide both sides of the equation by "v_trailer":
400 km ÷ v_trailer = 400 km ÷ v_bus + t.
Rearranging the equation:
400 km ÷ v_trailer - 400 km ÷ v_bus = t.
Now, let's express the ratio of the speed of the bus to that of the trailer:
r = v_bus ÷ v_trailer.
Dividing both sides of the equation by "v_trailer" and using the equation from earlier, we get:
r = v_bus ÷ v_trailer = (400 km ÷ v_bus + t) ÷ (400 km ÷ v_trailer).
Simplifying further:
r = (400 km ÷ v_bus + t) ÷ (400 km ÷ v_trailer).
Thus, the ratio of the speed of the bus to that of the trailer is (400 km ÷ v_bus + t) ÷ (400 km ÷ v_trailer).