Question

Катер прошел 20 км вверх по реке и 30 км вниз, затратив на весь путь 2 часа. Какова скорость течение реки, если скорость катера в стоячей воде 25 км/ч

Answer 1

X скорость течения
25+x скорость катера по течению
25-x скорость против течения
20/(25-x) время против течения
30/(25+x) время по течению
20/(25-x) + 30/(25+x)=2
20(25+x)+30(25-x)=2(25+x)(25-x)
500+20x+750-30x=2(25^2-x^2)
1250-10x=1250-2x^2
2x^2-10x=0
X^2-5x=0
X(x-5)=0
X=0 (не подходит)
Или
X-5=0
X=5
Ответ:5

Answer 2

Problem Analysis

We are given that a boat traveled 20 km upstream and 30 km downstream, taking a total of 2 hours for the entire journey. We are also given that the speed of the boat in still water is 25 km/h. We need to determine the speed of the river's current.

Solution

Let's assume the speed of the river's current is x km/h.

When the boat is traveling upstream (against the current), its effective speed is reduced by the speed of the current. Therefore, the boat's speed upstream is (25 - x) km/h.

When the boat is traveling downstream (with the current), its effective speed is increased by the speed of the current. Therefore, the boat's speed downstream is (25 + x) km/h.

We can use the formula time = distance / speed to calculate the time taken for each leg of the journey.

The time taken to travel 20 km upstream is given by: 20 / (25 - x) hours.

The time taken to travel 30 km downstream is given by: 30 / (25 + x) hours.

According to the problem statement, the total time taken for the entire journey is 2 hours. Therefore, we can write the equation:

20 / (25 - x) + 30 / (25 + x) = 2.

We can solve this equation to find the value of x.

Calculation

Let's solve the equation to find the value of x.

20 / (25 - x) + 30 / (25 + x) = 2

Multiplying through by the denominators, we get:

20(25 + x) + 30(25 - x) = 2(25 - x)(25 + x).

Expanding and simplifying, we get:

500 + 20x + 750 - 30x = 50^2 - x^2.

Combining like terms, we get:

20x - 30x + 750 + 500 = 2500 - x^2.

Simplifying further, we get:

-10x + 1250 = 2500 - x^2.

Rearranging the equation, we get:

x^2 - 10x + 1250 - 2500 = 0.

Simplifying, we get:

x^2 - 10x - 1250 = 0.

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

Using the quadratic formula, we have:

x = (-b ± √(b^2 - 4ac)) / (2a).

For our equation, a = 1, b = -10, and c = -1250.

Substituting the values, we get:

x = (-(-10) ± √((-10)^2 - 4(1)(-1250))) / (2(1)).

Simplifying, we get:

x = (10 ± √(100 + 5000)) / 2.

x = (10 ± √5100) / 2.

Calculating the square root of 5100, we get:

√5100 ≈ 71.42.

Substituting this value into the equation, we get:

x = (10 ± 71.42) / 2.

Simplifying further, we get two possible values for x:

x1 = (10 + 71.42) / 2 ≈ 40.71.

x2 = (10 - 71.42) / 2 ≈ -30.71.

Since the speed of the river's current cannot be negative, we discard the negative value.

Therefore, the speed of the river's current is approximately 40.71 km/h.

Answer

The speed of the river's current is approximately 40.71 km/h.