Question

Два мотоцикла стартуют одновременно в одном направлении из двух диаметрально противоположных точек круговой трассы, длина которой равна 20 км. Через сколько минут мотоциклы поравняются в первый раз, если скорость одного из них на 12 км/ч больше скорости другого?

Answer 1

Пусть х км/ч - скорость первого мотоциклиста, (х+12) км/ч - скорость второго мотоциклиста,
общая скорость движения (скорость сближения)  (х+12)-х=12 км/ч,  
расстояние между ними 10 км.
Через 10:12=5/6 час= 50 мин мотоциклы поравняются в первый раз.

Answer 2

Problem Analysis

We have two motorcycles starting simultaneously from diametrically opposite points on a circular track with a length of 20 km. One motorcycle is traveling 12 km/h faster than the other. We need to determine how many minutes it will take for the motorcycles to meet for the first time.

Solution

To solve this problem, we can use the concept of relative speed. The relative speed between the two motorcycles is the difference in their speeds. Let's assume the speed of the slower motorcycle is x km/h. Therefore, the speed of the faster motorcycle will be x + 12 km/h.

To find the time it takes for the motorcycles to meet, we need to determine the distance they need to cover to meet for the first time. Since they are starting from diametrically opposite points, they will meet when they have covered half the circumference of the circular track.

The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle. In this case, the radius is half the length of the circular track, which is 20/2 = 10 km.

Therefore, the distance the motorcycles need to cover to meet for the first time is 10 km.

Now, we can use the formula time = distance / speed to find the time it takes for the motorcycles to meet.

For the slower motorcycle: time = 10 km / x km/h

For the faster motorcycle: time = 10 km / (x + 12) km/h

Since the motorcycles are starting simultaneously, the time it takes for them to meet will be the same. Therefore, we can set up the following equation:

10 / x = 10 / (x + 12)

To solve this equation, we can cross-multiply:

10(x + 12) = 10x

Simplifying the equation:

10x + 120 = 10x

We can see that the 10x terms cancel out, leaving us with:

120 = 0

This equation has no solution, which means the motorcycles will never meet on the circular track.

Conclusion

The motorcycles will never meet on the circular track, regardless of their speeds.