Question

Катер прошел по течению 36 км, развернулся и прошел еще 48 км против течения, скорость течения 3 км/ч, всего катер был в пути 6 часов, найдите скорость катера в стоячей воде

Answer 1

Пусть скорость катера в стоячей воде будет х км/ч, тогда скорость по течению равна (x+3) км/ч, а против течения - (x-3) км/ч. Время, затраченное по течению - $\dfrac{36}{x+3}$ ч, а против течения - $\dfrac{48}{x-3}$. На весь путь катер затратил $\dfrac{36}{x+3} + \dfrac{48}{x-3}$. Зная, что катер прошел 3 часа. Составим уравнение

                                 $\dfrac{36}{x+3} + \dfrac{48}{x-3} =6\\ \\ x^2-14x-15=0.$

По т. Виета:
 $x_1=-1$ - не удовлетворяет условию;
 $x_2=15$ км/ч - скорость катера в стоячей воде.

Ответ: 15 км/ч.

Answer 2

Calculation of Boat Speed in Still Water

To find the speed of the boat in still water, we can use the information provided:

- The boat traveled 36 km downstream (with the current) and then turned around and traveled 48 km upstream (against the current). - The speed of the current is 3 km/h. - The total time the boat was in motion is 6 hours.

Let's calculate the speed of the boat in still water step by step:

1. Let's assume the speed of the boat in still water is x km/h. 2. When the boat is traveling downstream, its effective speed is the sum of its speed in still water and the speed of the current. Therefore, the effective speed downstream is (x + 3) km/h. 3. The time taken to travel downstream is 36 km / (x + 3) km/h. 4. When the boat is traveling upstream, its effective speed is the difference between its speed in still water and the speed of the current. Therefore, the effective speed upstream is (x - 3) km/h. 5. The time taken to travel upstream is 48 km / (x - 3) km/h. 6. The total time taken for the entire journey is 6 hours. 7. Therefore, the equation can be formed as follows:

36 / (x + 3) + 48 / (x - 3) = 6

To solve this equation and find the value of x, we can simplify it:

1. Multiply both sides of the equation by (x + 3)(x - 3) to eliminate the denominators. 2. Simplify the equation and bring all the terms to one side. 3. Solve the resulting quadratic equation to find the value of x.

Please note that the calculation of the exact value of x requires solving the quadratic equation, which can be a bit complex. However, I can provide you with the simplified equation and the approximate value of x.

Let's proceed with the calculation:

Calculation Steps:

1. Multiply both sides of the equation by (x + 3)(x - 3):

36(x - 3) + 48(x + 3) = 6(x + 3)(x - 3)

2. Simplify the equation:

36x - 108 + 48x + 144 = 6(x^2 - 9)

84x + 36 = 6x^2 - 54

6x^2 - 84x - 90 = 0

3. Solve the quadratic equation. Using the quadratic formula, we get:

x = (-(-84) ± √((-84)^2 - 4 * 6 * (-90))) / (2 * 6)

Simplifying further:

x = (84 ± √(7056 + 2160)) / 12

x = (84 ± √9216) / 12

x = (84 ± 96) / 12

x = (84 + 96) / 12 or x = (84 - 96) / 12

x = 180 / 12 or x = -12 / 12

x = 15 or x = -1

Since the speed of the boat cannot be negative, the speed of the boat in still water is 15 km/h.

Please note that this calculation assumes a linear relationship between the speed of the boat and the distance traveled. In reality, factors such as water conditions and boat design can affect the relationship between speed and distance.