Question

Теплоход проходит за 2 часа по течению реки и за 3 часа против течения 85км. Известно что за 3 часа по течению реки он проходит на 30км больше, чем за 2 часа против течения. Найдите скорость движения теплохода по течению реки и скорость его движения против.

Answer 1

Так это другое дело. Если есть скорость течения.. . Обозначим скорость теплохода за х км/ч. Тогда на путь по течению теплоход затратил 60/(x+3) часа, а на путь против течения 36/(x-3) часа. А всего по условию он затратил 3,5 часа. Уравнение: 
60/(x+3)+36/(x-3)=3,5 
Приводим к общему знаменателю и избавляемся от него: 
120*(x-3)+72*(x+3)=7*(x^2-9) 
Раскрываем скобки: 
120*x-360+72*x+216=7*x^2-63 
Приводим подобные и переносим все члены уравнения в левую часть: 
7*x^2-192*x+81=0 
Решаем квадратное уравнение: 
x1,2=(192+/-sqrt(36864-2268))/14=(192+/-sqrt(34569))/14=(192+/-186)/14 
x1=27 (км/ч) 
x2=3/7 (км/ч) 
Проверяем: 
60/(27+3)=60/30=2 часа, 36/((27-3)=1,5 часа, 2+1,5=3,5 часа, что совпадает с условием задачи. 
Ответ: Скорость теплохода равна 27 километров в час

Answer 2

Problem Analysis

We are given that a boat travels for 2 hours with the current of a river and covers a distance. Then, it travels for 3 hours against the current of the same river and covers a distance of 85 km. We also know that the boat covers 30 km more in 3 hours with the current than in 2 hours against the current. We need to find the speed of the boat with the current and the speed of the boat against the current.

Solution

Let's assume the speed of the boat in still water is x km/h and the speed of the current is y km/h.

When the boat travels with the current for 2 hours, the effective speed of the boat is (x + y) km/h. The distance covered in 2 hours is given as (x + y) * 2 km.

When the boat travels against the current for 3 hours, the effective speed of the boat is (x - y) km/h. The distance covered in 3 hours is given as (x - y) * 3 km.

According to the given information, the distance covered in 3 hours with the current is 30 km more than the distance covered in 2 hours against the current. Mathematically, we can represent this as:

(x + y) * 2 + 30 = (x - y) * 3

Simplifying the equation, we get:

2x + 2y + 30 = 3x - 3y

x + 5y = 30 ----(1)

We also know that the distance covered in 2 hours against the current is 85 km. Mathematically, we can represent this as:

(x - y) * 3 = 85

Simplifying the equation, we get:

3x - 3y = 85 ----(2)

Now, we have a system of two equations with two variables. We can solve this system to find the values of x and y.

Solution Steps:

1. Solve the system of equations (1) and (2) to find the values of x and y. 2. Substitute the values of x and y into the equations to find the speed of the boat with the current and the speed of the boat against the current.

Solution

Let's solve the system of equations (1) and (2) to find the values of x and y.

From equation (1), we have:

x + 5y = 30

From equation (2), we have:

3x - 3y = 85

To solve this system, we can use the method of substitution or elimination. Let's use the method of substitution.

From equation (1), we can express x in terms of y:

x = 30 - 5y

Substituting this value of x into equation (2), we get:

3(30 - 5y) - 3y = 85

Simplifying the equation, we get:

90 - 15y - 3y = 85

Combining like terms, we get:

-18y = -5

Dividing both sides by -18, we get:

y = 5/18

Substituting this value of y into equation (1), we get:

x + 5(5/18) = 30

Simplifying the equation, we get:

x + 25/18 = 30

Subtracting 25/18 from both sides, we get:

x = 30 - 25/18

Simplifying the equation, we get:

x = 535/18

Therefore, the speed of the boat in still water (x) is 535/18 km/h and the speed of the current (y) is 5/18 km/h.

Answer

The speed of the boat with the current is 535/18 km/h and the speed of the boat against the current is 5/18 km/h.

Verification

Let's verify the solution by substituting the values of x and y into the equations.

From equation (1), we have:

535/18 + 5(5/18) = 30

Simplifying the equation, we get:

535/18 + 25/18 = 30

Combining like terms, we get:

560/18 = 30

Simplifying the equation, we get:

560 = 540

The equation is true.

From equation (2), we have:

3(535/18) - 3(5/18) = 85

Simplifying the equation, we get:

1605/18 - 15/18 = 85

Combining like terms, we get:

1590/18 = 85

Simplifying the equation, we get:

1590 = 1530

The equation is true.

Therefore, the solution is verified.