Question

катер отправился в путь в 12 часов дня, прошел 11 км по течению реки и сделал остановку на 2 часа. после этого он прошел еще 27 км против течения и прибыл в пункт назначения в 16 часов. найдите собственную скорость катера, если скорость течения реки 2 км/ч. помогите пожалуйста сейчас прям надо!!!!)))

Answer 1

Problem Analysis

We are given the following information: - A boat set off at 12:00 PM. - It traveled 11 km downstream (with the current) and made a 2-hour stop. - After the stop, it traveled 27 km upstream (against the current) and arrived at its destination at 4:00 PM. - The speed of the river's current is 2 km/h.

We need to find the speed of the boat.

Solution

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

When the boat is traveling downstream, its effective speed is the sum of its own speed and the speed of the current. Therefore, the effective speed downstream is (x + 2) km/h.

When the boat is traveling upstream, its effective speed is the difference between its own speed and the speed of the current. Therefore, the effective speed upstream is (x - 2) km/h.

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

Let's calculate the time taken for each leg of the journey:

- Time taken downstream: 11 km / (x + 2) km/h. - Time taken upstream: 27 km / (x - 2) km/h.

The total time taken for the journey is the sum of the time taken downstream, the 2-hour stop, and the time taken upstream. This should be equal to 4 hours:

11 / (x + 2) + 2 + 27 / (x - 2) = 4.

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

Let's solve the equation step by step:

11 / (x + 2) + 2 + 27 / (x - 2) = 4

11 / (x + 2) + 27 / (x - 2) = 2

To simplify the equation, let's multiply both sides by (x + 2)(x - 2):

11(x - 2) + 27(x + 2) = 2(x + 2)(x - 2)

Simplifying further:

11x - 22 + 27x + 54 = 2(x^2 - 4)

38x + 32 = 2x^2 - 8

Rearranging the equation:

2x^2 - 38x - 40 = 0

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

Using the quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / (2a), where a = 2, b = -38, and c = -40.

Calculating the discriminant: b^2 - 4ac = (-38)^2 - 4(2)(-40) = 1444 + 320 = 1764.

Taking the square root of the discriminant: sqrt(1764) = 42.

Now, substituting the values into the quadratic formula:

x = (-(-38) ± 42) / (2(2))

Simplifying further:

x = (38 ± 42) / 4

We have two possible solutions:

1. x = (38 + 42) / 4 = 80 / 4 = 20 2. x = (38 - 42) / 4 = -4 / 4 = -1

Since the speed of the boat cannot be negative, the speed of the boat is 20 km/h.

Therefore, the speed of the boat is 20 km/h.

Answer

The speed of the boat is 20 km/h.