Question

Четырех школьников, которые бегают с разной скоростью, разбили на две команды по два человека, и в первую включили самого быстрого из них. Вначале команды соревновались в том, бегуны какой команды раньше встретятся, если на дистанции в 400м они побегут навстречу друг другу. Оказалось, что командам потребовалось одинаковое время. А какая команда быстрее пробежит эстафету 2 × 400 м? (В эстафете каждый спортсмен пробегает 400 метров. Считается, что скорость бегунов во время забегов постоянна.)

Answer 1

Скорость самого быстрого - n1
Скорость его партнера по команде - n2
Скорость члена второй команды - n3
Скорость его партнера по команде - n4
Рассмотрим ситуацию, когда ребята из первой команды бегут друг другу навстречу: скорость сближения ребят(1 ком.) = n1 + n2; (2 ком.) = n3 + n4
Раз дистанция одинаковая, то из условия n1 + n2 = n3 + n4
Теперь 2x400:
Время, которой затратит n1 = 400/n1
Второй затратит 400/n2
Третий 400/n3
И, наконец, четвертый 400/n4
Сложим скорости партнеров по команде: (400(n1+n2))/(n1*n2) и (400(n3+n4))/(n1*n2); Сумма n1 + n2 и n3 + n4 одинакова, поэтому смотрим на n1*n2 и n3*n4; Очевидно, что победит тот, у кого время окажется меньше, а значит победит тот, у кого произведение скоростей больше.
Так как скорости у всех разные, и в первой команде есть самый быстрый, то очевидно, что значение n3 или n4 находится между значениями n1 и n2. Теперь представим n1*n2 и n3*n4 как прямоугольники и наложим их друг на друга(нарисуй на листе). В итоге получается один прямоугольник(общий) со сторонами a<n1 и b>n2, но остались и маленькие узкие прямоугольники; Для простоты возьмем n1 = 9; n2 = 7, тогда пусть a = 8,5; b = 7,5; Теперь найдем площадь всей фигуры: 7*8,5 + 0,5*8,5 +0,5*7,5 = 8,5(7 + 0,5) + 0,5*7,5 = 8,5*7,5 + 0,5*7,5 = 7,5(8,5 + 0,5) = 7,5*9; Теперь вычтем 9*7, очевидно, что 7,5*9>9*7⇒n3*n4>n1*n2⇒вторая команда быстрее пробежит эстафету 2x400 м.

Answer 2

Problem Analysis

We have four schoolchildren who run at different speeds. They are divided into two teams, each consisting of two members. The first team includes the fastest runner. The teams compete to see which team's runners will meet first if they run towards each other on a 400-meter track. The teams took the same amount of time to meet. We need to determine which team will complete a 2x400-meter relay faster.

Solution

To determine which team will complete the 2x400-meter relay faster, we need to consider the relative speeds of the runners in each team.

Let's assume the four runners are labeled A, B, C, and D, with A being the fastest runner.

The first team consists of the fastest runner (A) and another runner (B). The second team consists of the remaining two runners (C and D).

Since the teams took the same amount of time to meet while running towards each other on a 400-meter track, we can conclude that the total distance covered by the first team (A and B) is equal to the total distance covered by the second team (C and D).

Since the runners maintain a constant speed during the race, we can assume that the ratio of the speeds of the runners in each team is the same as the ratio of the distances covered by each team.

Let's denote the speeds of the runners in the first team as v1 and v2, and the speeds of the runners in the second team as v3 and v4.

We can set up the following equation based on the distances covered by each team:

Distance covered by the first team (A and B) = Distance covered by the second team (C and D)

(2 * 400) / (v1 + v2) = (2 * 400) / (v3 + v4)

Simplifying the equation, we get:

v1 + v2 = v3 + v4

Since the first team consists of the fastest runner (A) and another runner (B), we can assume that v1 > v2. Similarly, since the second team consists of the remaining two runners (C and D), we can assume that v3 > v4.

Based on these assumptions, we can conclude that the first team (A and B) will complete the 2x400-meter relay faster than the second team (C and D).

Therefore, the first team is faster in the relay.

Example Calculation

Let's consider an example to illustrate the solution.

Assume the speeds of the runners are as follows: - Runner A: 10 m/s - Runner B: 8 m/s - Runner C: 7 m/s - Runner D: 6 m/s

Using the equation v1 + v2 = v3 + v4, we can substitute the values:

10 + 8 = 7 + 6

18 = 13

Since the equation is not balanced, we can conclude that our assumption was incorrect. In this case, the second team (C and D) would be faster in the relay.

Please note that this is just an example calculation, and the actual speeds of the runners are not provided in the question. The solution is based on the assumption that the speeds of the runners are known.

Conclusion

Based on the given information, we can conclude that the first team (consisting of the fastest runner and another runner) will complete the 2x400-meter relay faster than the second team. However, without specific information about the speeds of the runners, we cannot provide a definitive answer.