Question

Идя в школу , ученик заметил , что каждые 6 мин его настигает автобус , а каждые 3 мин мимо проходит встречный автобус. Через интервалы времени отходят автобусы от своих конечных остановок , если известно , что они уходят через одинаковые промежутки времени? Скорость автобусов на всем маршруте условно считать постоянной.

Answer 1

Т - интервал отхода автобуса
v - скорость автобуса
u - скорость пешехода
пусть автобус догнал пешехода, следующий автобус в этой точке будет через время T. за єто время пешеход прошел вперед на u*T. автобус догонит его за время u*T/(v-u)
интервал через сколько автобусы обганяют пешеходов равен t1=T+u*T/(v-u)=T*v/(v-u)
интервал через сколько автобусы встречают пешеходов равен t2=T-u*T/(v+u)=T*v/(v+u)
*************
t1=T*v/(v-u)
t2=T*v/(v+u)
**************
1/t1=(v-u)/(T*v)
1/t2=(v+u)/(T*v)
**************
1/t1+1/t2=(v-u)/(T*v)+(v+u)/(T*v)=2/T
**************
T=2/(1/t1+1/t2)=2*t1*t2/(t1+t2)=2*6*3/(6+3) мин = 4 мин

Answer 2

Problem Analysis

A student walking to school notices that a bus catches up to them every 6 minutes, while a bus going in the opposite direction passes by every 3 minutes. The buses depart from their respective end stops at regular intervals. The question is, if the intervals are the same, what is the speed of the buses along the entire route?

Solution

To find the speed of the buses, we need to determine the time it takes for each bus to travel the distance between the end stops. Let's assume the distance between the end stops is d.

Let's consider the time it takes for the first bus to travel the distance d. Since the bus catches up to the student every 6 minutes, we can say that in 6 minutes, the bus travels the distance d. Therefore, the speed of the first bus is d/6.

Similarly, let's consider the time it takes for the second bus to travel the distance d. Since the second bus passes by every 3 minutes, we can say that in 3 minutes, the bus travels the distance d. Therefore, the speed of the second bus is d/3.

Since the speed of each bus is equal to the distance divided by the time, we can equate the speeds of the two buses:

d/6 = d/3

Simplifying the equation, we find:

d = 2d/6

Multiplying both sides by 6, we get:

6d = 2d

Dividing both sides by 2d, we find:

6 = 2

This equation is not true, which means there is no consistent solution for the speed of the buses along the entire route. Therefore, we cannot determine the speed of the buses based on the given information.

Conclusion

Based on the given information, we cannot determine the speed of the buses along the entire route. The equation derived from the information leads to an inconsistency, indicating that there is no consistent solution for the speed of the buses.

Answer 3

Problem Analysis

A student notices that a bus catches up to him every 6 minutes while he is walking to school, and a bus going in the opposite direction passes by every 3 minutes. The buses depart from their respective endpoints at regular intervals. The question is whether the buses depart at the same interval.

Solution

To determine if the buses depart at the same interval, we need to find the least common multiple (LCM) of 6 and 3. The LCM will give us the time it takes for both buses to depart simultaneously.

Finding the LCM

The LCM of two numbers can be found by dividing their product by their greatest common divisor (GCD). In this case, the product of 6 and 3 is 18, and the GCD is 3. Therefore, the LCM is 18 divided by 3, which is 6 minutes.

Conclusion

The buses depart from their respective endpoints at the same interval of 6 minutes. This means that every 6 minutes, both buses will depart simultaneously.

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