Question

Караван идет со скоростью 6 км/ч из города А в город В, расстояние между которыми равно 42 км. Прибыв в город В, караван после часовой стоянки отправился в направлении, перпендикулярном АВ. Через 8 часов после отправления каравана из города А для встречи с ним оттуда выехал гонец со скоростью 10 км/ч по кратчайшему пути АС. За какое наименьшее время часов гонец сможет догнать караван? Ответ запишите в минутах.

Answer 1

С учетом отдыха движение началось одновременно.
Смотрим на схему и видим прямоугольный треугольник и можно применить теорему Пифагора 
 42² +(6t)² = (10*t)² или
(100-36)*t² = (8*t)² = 42² или 
8*t = 42 или
t=  42/8 = 5.25 часа = 5,25*60 =315 мин - ОТВЕТ
Дополнительно
Пути  6 км/ч *5,25 ч = 31,5 км и гонец -10 км/ч *5,25 ч  = 52,5 км

Answer 2

Problem Analysis

We are given that a caravan is traveling from city A to city B at a speed of 6 km/h, with a distance of 42 km between the two cities. After reaching city B, the caravan takes an hour-long break and then continues its journey in a direction perpendicular to AB. After 8 hours of the caravan's departure from city A, a messenger leaves city A to meet the caravan, traveling at a speed of 10 km/h along the shortest path AC. We need to determine the minimum time, in minutes, it will take for the messenger to catch up with the caravan.

Solution

To solve this problem, we can use the concept of relative velocity. The messenger's speed is 10 km/h, while the caravan's speed is 6 km/h. Since the messenger is traveling in a perpendicular direction to AB, we can consider the distance traveled by the messenger as the hypotenuse of a right-angled triangle, with the sides representing the distances traveled by the messenger and the caravan.

Let's calculate the distance traveled by the messenger after 8 hours: Distance traveled by the messenger = Speed * Time = 10 km/h * 8 h = 80 km

Now, let's calculate the distance between the messenger and the caravan at this point. Since the messenger is traveling along the shortest path AC, which is perpendicular to AB, we can consider the distance between the messenger and the caravan as the altitude of the right-angled triangle formed by the messenger, the caravan, and the line AB.

Using the Pythagorean theorem, we can calculate the altitude: Altitude = √(Hypotenuse^2 - Base^2) = √(80^2 - 42^2) = √(6400 - 1764) = √(4636) ≈ 68.09 km

Now, we can calculate the time it will take for the messenger to catch up with the caravan. Since the messenger's speed is 10 km/h and the distance between them is 68.09 km, we can use the formula: Time = Distance / Speed = 68.09 km / 10 km/h = 6.809 hours

Finally, we need to convert the time to minutes: Time in minutes = Time * 60 = 6.809 hours * 60 = 408.54 minutes

Therefore, the messenger will be able to catch up with the caravan in approximately 408.54 minutes.

Answer

The messenger will be able to catch up with the caravan in approximately 408.54 minutes.