Question

Папа и сын плывут на лодке против течения. В какой-то момент сын уронил за борт папину шляпу. Только через 15 минут папа заметил пропажу, быстро развернул лодку, и они поплыли по течению с той же собственной скоростью. За сколько минут они догонят шляпу? ПРОСЬБА: не копировать из интернета!!!

Answer 1

Через 15 минут.
Обозначим: скорость реки Р, скорость гребцов Г.
 В точке А сын уронил за борт папину шляпу. Только через 15 минут папа заметил пропажу , быстро развернул лодку в точке В (за это время шляпа проплыла до точки Б путь АБ=15*Р, гребцы на лодке - до точки В путь АВ=15*(Г-Р) и они поплыли по течению с той же СОБСТВЕННОЙ скоростью (гребли ОДИНАКОВО).
 
Лодка двигалась ПО ТЕЧЕНИЮ, скорости гребцов и реки СКЛАДЫВАЛИСЬ, поэтому за время Х (до встречи со шляпой в точке Д) лодкой пройден путь ВД=Х*(Р+Г),  шляпой -  путь БД=Х*Р.

Нарисуйте себе картинку с точками и увидите, что ВД=ВА+АБ+БД
Осталось подставить Х*(Р+Г)=15*(Г-Р) + 15*Р + Х*Р
Раскрываем скобки ХР+ХГ=15Г-15Р+15Р+ХР
ХГ=15Г
Х=15

Answer 2

Problem Analysis

In this problem, a father and son are traveling in a boat against the current. At some point, the son accidentally drops the father's hat overboard. The father notices the loss after 15 minutes and quickly turns the boat around. They then start traveling downstream at the same speed as before. The question is, how long will it take for them to catch up to the hat?

Solution

To solve this problem, we need to consider the relative speeds of the boat and the hat.

Let's assume that the speed of the boat relative to the water is x units per minute, and the speed of the current is y units per minute.

When the boat is traveling against the current, its effective speed is reduced by the speed of the current. So the boat's speed against the current is x - y units per minute.

When the boat turns around and travels downstream, its effective speed is increased by the speed of the current. So the boat's speed downstream is x + y units per minute.

Since the boat and the hat are both traveling downstream, their speeds are the same. Therefore, the time it takes for the boat to catch up to the hat is equal to the time it takes for the hat to travel the distance it was dropped.

Let's denote the distance the hat was dropped as d units. The time it takes for the hat to travel this distance is given by the formula:

time = distance / speed

Substituting the values, we have:

time = d / (x + y)

Therefore, the boat will catch up to the hat in d / (x + y) minutes.

Answer

The boat will catch up to the hat in d / (x + y) minutes.

Please note that the specific values for the speed of the boat relative to the water (x) and the speed of the current (y) were not provided in the question. Without these values, we cannot determine the exact time it will take for the boat to catch up to the hat.