Question

Папа, мама и сын стартовали одновременно и ездят в одну сторону с постоянными скоростями по круговой велодорожке. Папа проезжает круг за полторы минуты, мама — за три с половиной минуты, сын — за восемь минут. У них есть на всех одна фляжка с водой, на старте она была у сына. Если кто-то с фляжкой обгоняет другого или его обгоняет другой, то фляжка передается другому. Езда прекратилась, когда все они одновременно снова оказались на линии старта. Сколько кругов сделала фляжка?

Answer 1

Нужно построить 3 синусоиды на одном графике:

Все три синусоиды ОДНОВРЕМЕННО пересекают ось ОХ  в точке 66.

За это время папа проедет 44 круга, мама 33, сын 6 кругов.

По тому же графику можно посчитать количество кругов, сделанных фляжкой (пересечение любой синусоиды с ОХ - линия старта).

(я насчитал 30).

Answer 2

Problem Analysis

We are given that a father, mother, and son start simultaneously and ride in the same direction on a circular bike path. Each of them has a constant speed, and we are given the time it takes for each of them to complete one lap around the path. The water bottle is initially with the son, and if someone with the water bottle overtakes another person or gets overtaken, they pass the water bottle to the other person. The ride ends when all three of them simultaneously reach the starting line again. We need to determine how many laps the water bottle made during the ride.

Solution

To solve this problem, we need to find the least common multiple (LCM) of the time it takes for each person to complete one lap. The LCM will give us the time it takes for all three of them to simultaneously reach the starting line again. Since the water bottle starts with the son, we need to find the number of laps the son completes during this time.

Let's calculate the LCM of the given lap times:

- The father completes one lap in 1.5 minutes. - The mother completes one lap in 3.5 minutes. - The son completes one lap in 8 minutes.

To find the LCM, we can use the prime factorization method or the division method. Let's use the division method:

1. Write the given lap times in the form of prime factors:

- Father's lap time: 1.5 minutes = 3 * 0.5 minutes - Mother's lap time: 3.5 minutes = 7 * 0.5 minutes - Son's lap time: 8 minutes = 2^3 minutes

2. Find the highest power of each prime factor that appears in any of the given lap times:

- Prime factor 2 appears with the highest power of 3. - Prime factor 3 appears with the highest power of 1. - Prime factor 7 appears with the highest power of 1.

3. Multiply these prime factors with their highest powers:

LCM = 2^3 * 3^1 * 7^1 = 8 * 3 * 7 = 168 minutes.

Therefore, it takes 168 minutes for all three of them to simultaneously reach the starting line again.

Now, let's find the number of laps the son completes during this time. Since the son completes one lap in 8 minutes, we can divide the total time (168 minutes) by the son's lap time:

Number of laps = 168 minutes / 8 minutes = 21 laps.

So, the water bottle completes 21 laps during the ride.

Answer

The water bottle completes 21 laps during the ride.