Question

Из двух городов А и В одновременно навстречу друг другу с постоянными скоростями выехали два автомобиля. Первый автомобиль приехал в город в через 16 часов после встречи, а второй в город А через 25 часов после встречи. За какое время первый проезжает путь от А до В?

Answer 1

Пусть время прошедшее до момента встречи равно t (в момент встречи оба автомобиля были в пути одинаковое количество часов t), тогда все время в пути:
Для первого: t+16
Для второго: t+25

Расстояние между городами примем равным L, тогда выразим скорости:
Для первого: L:(t+16)
Для второго: L:(t+25)

Т.к. в момент встречи автомобили проехали все расстояние, то скорость их сближения: L:t

Скорость сближения это сумма скоростей, тогда получим уравнение:
$\frac{L}{t}= \frac{L}{t+16}+ \frac{L}{t+25}$
Избавимся от L разделив обе части на L:
$\frac{1}{t}= \frac{1}{t+16}+ \frac{1}{t+25}\\ \\ \frac{1}{t}= \frac{t+25}{(t+16)(t+25)}+ \frac{t+16}{(t+25)(t+16)}\\ \\ \frac{1}{t}= \frac{t+25+t+16}{(t+16)(t+25)}\\ \\ \frac{1}{t}= \frac{2t+41}{t^2+41t+400}\\ \\ 1*(t^2+41t+400)=t*(2t+41)\\ t^2+41t+400=2t^2+41t\\ 400=t^2\\ t= \sqrt{400} \\ t=20$

Тогда первый проезжает путь от А до В за 20+16=36 часов
ОТВЕТ 36

Answer 2

Problem Analysis

We have two cities, A and B, and two cars that start simultaneously from each city and travel towards each other with constant speeds. The first car arrives in city B 16 hours after the meeting, while the second car arrives in city A 25 hours after the meeting. We need to determine the time it takes for the first car to travel from city A to city B.

Solution

Let's assume the speed of the first car is v1 and the speed of the second car is v2. We need to find the time it takes for the first car to travel from city A to city B.

To solve this problem, we can use the formula: time = distance / speed.

Let's assume the distance between city A and city B is d. Since the two cars meet somewhere between the two cities, the total distance traveled by both cars is equal to the distance between the two cities.

The time it takes for the first car to travel from city A to the meeting point is 16 hours. The time it takes for the second car to travel from city B to the meeting point is 25 hours.

Using the formula, we can write two equations:

1. For the first car: 16 = (d - x) / v1, where x is the distance traveled by the first car from the meeting point to city B. 2. For the second car: 25 = (d - x) / v2, where x is the distance traveled by the first car from the meeting point to city B.

We can solve these equations to find the value of x and then calculate the time it takes for the first car to travel from city A to city B.

Let's solve the equations:

From equation 1: 16 = (d - x) / v1, we can rewrite it as: 16v1 = d - x.

From equation 2: 25 = (d - x) / v2, we can rewrite it as: 25v2 = d - x.

Since both expressions are equal to d - x, we can equate them:

16v1 = 25v2.

Now, we can solve this equation to find the value of x.

Calculation

Let's calculate the value of x using the given information.

From the equation 16v1 = 25v2, we can rearrange it to solve for x:

x = d - 16v1.

Substituting this value of x into the equation 25v2 = d - x, we get:

25v2 = d - (d - 16v1).

Simplifying the equation, we have:

25v2 = 16v1.

Now, we can solve this equation to find the value of v1 in terms of v2:

v1 = (25v2) / 16.

Since we know the value of v1, we can substitute it back into the equation x = d - 16v1 to find the value of x:

x = d - 16((25v2) / 16).

Simplifying the equation, we have:

x = d - 25v2.

Now, we can substitute the value of x into the equation 16v1 = d - x to find the value of d:

16v1 = d - (d - 25v2).

Simplifying the equation, we have:

16v1 = 25v2.

Now, we can solve this equation to find the value of d:

d = (16v1) / (1 - 25/16).

Substituting the given values of v1 and v2 into the equation, we get:

d = (16 * v1) / (1 - 25/16).

Now, we can calculate the value of d:

d = (16 * v1) / (1 - 25/16).

Using the calculated value of d, we can find the time it takes for the first car to travel from city A to city B:

time = distance / speed.

Substituting the values of d and v1 into the equation, we get:

time = d / v1.

Now, we can calculate the time it takes for the first car to travel from city A to city B.

Answer

The time it takes for the first car to travel from city A to city B is (16 * v1) / (1 - 25/16) hours.

Please note that the actual values of v1 and v2 are not provided in the question, so we cannot calculate the exact time. However, the formula provided above will give you the time in terms of the speeds of the two cars.