Question

Петя и Вася ездят на велосипедах по кругу с постоянными скоростями. Скорость Пети равна 8 8 км/ч, а скорость Васи — — 18 18 км/ч. Вначале они ехали в разные стороны (Петя — — по часовой стрелке, а Вася — — против), а затем Петя изменил направление движения (начал двигаться против часовой стрелки) и одновременно увеличил свою скорость вдвое. После этого велосипедисты стали встречаться в k раз реже. Найдите k .

Answer 1

Ответ: в 6 раз реже

Answer 2

Problem Analysis

In this problem, we are given that Petya and Vasya are riding bicycles in a circular path at constant speeds. Petya's speed is 8 km/h, while Vasya's speed is 18 km/h. Initially, they were riding in opposite directions, with Petya moving clockwise and Vasya moving counterclockwise. After some time, Petya changes his direction and starts moving counterclockwise, while simultaneously doubling his speed. We need to find the value of k, which represents how many times less frequently the cyclists meet after Petya changes his direction and doubles his speed.

Solution

Let's assume that the circumference of the circular path is C km. Since Petya's speed is 8 km/h, he covers a distance of 8 km in 1 hour. Similarly, Vasya covers a distance of 18 km in 1 hour. Therefore, the time taken by Petya and Vasya to meet initially is given by:

Time taken to meet initially = C / (8 + 18) = C / 26

After Petya changes his direction and doubles his speed, his new speed becomes 2 * 8 = 16 km/h. Now, the time taken by Petya and Vasya to meet is given by:

Time taken to meet after Petya changes his direction = C / (16 + 18) = C / 34

We are given that the cyclists meet k times less frequently after Petya changes his direction and doubles his speed. This means that the new time taken to meet is k times greater than the initial time taken to meet. Therefore, we can write the equation:

C / 34 = k * (C / 26)

Simplifying the equation, we get:

k = 34 / 26 = 17 / 13 ≈ 1.308

Therefore, the value of k is approximately 1.308.

Answer

The value of k is approximately 1.308.