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
(t +1) ¾ = 400 + t
(b) Find the ration of the speed of the bus so that of the trailer.
To solve this problem, let's break it down step by step.
(a) Express the average speed of the trailer in terms of t:
First, let's find the time it took for the bus to reach Q. Since the distance between P and Q is 400 km and the bus traveled the whole distance before returning, we can calculate the time as follows:
Speed = Distance/Time
Time = Distance/Speed
Time = 400 km / Speed of the bus
Now, let's consider the time taken for the bus to return from Q to the halfway point between P and Q. Since the total distance is 400 km, the halfway point is at 200 km. So, the bus has traveled a distance of 200 km when it meets the trailer.
The time taken for the bus to reach the halfway point is given as (t + 1).
Next, let's consider the time taken for the trailer to travel from P to the halfway point (200 km). We are given that they meet t hours after the bus leaves P. Thus, the time taken for the trailer is t hours.
We know that the average speed is given by the formula:
Average Speed = Total Distance / Total Time
Total Distance = Distance from P to halfway point + Distance from halfway point to P
Total Distance = 200 km + 200 km = 400 km
Total Time = Time taken by the bus to reach Q + Time taken by the bus to reach the halfway point + Time taken by the trailer to reach halfway point
Total Time = (400 km / Speed of the bus) + (t + 1) + t
Using the formula for average speed, we can express the average speed of the trailer in terms of t:
Average Speed of the trailer = Total Distance / Total Time
Average Speed of the trailer = 400 km / ((400 km / Speed of the bus) + (t + 1) + t)
Therefore, the average speed of the trailer in terms of t is:
Average Speed of the trailer = 400 / ((400 / S) + (2t + 1))
(b) Find the ratio of the speed of the bus to that of the trailer:
To find the ratio, we divide the speed of the bus by the speed of the trailer:
Ratio = Speed of the bus / Speed of the trailer
Ratio = S / (400 / ((400 / S) + (2t + 1)))
Therefore, the ratio of the speed of the bus to that of the trailer is S divided by (400 / ((400 / S) + (2t + 1))).