Question

Помогите пожалуйста решить эту задачу до 05.02.2014Лодка вышла из пункта А в 08.00, и, пройдя вниз по течению реки 36 км, прибыла в пункт B. Сделав там остановку на 1 ч, лодка вернулась обратно в пункт А в 14.00 того же дня. Определите собственную скорость лодки, если скорость течения реки равна 3 км/ч.

Answer 1

Пусть х - скорость лодки, тогда
36\(x+3) + 36\(x-3) = 6 часов - 1 час остановки

Answer 2

Problem Analysis

To solve this problem, we need to determine the speed of the boat. We are given that the boat traveled downstream for 36 km and then returned to the starting point. We also know that the speed of the river's current is 3 km/h.

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 + 3) 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 - 3) km/h.

We can use the formula distance = speed × time to calculate the time it takes for the boat to travel downstream and upstream.

1. Downstream journey: - Distance = 36 km - Speed = (x + 3) km/h - Time = Distance / Speed = 36 / (x + 3) hours

2. Upstream journey: - Distance = 36 km - Speed = (x - 3) km/h - Time = Distance / Speed = 36 / (x - 3) hours

We are given that the boat left point A at 08:00 and returned to point A at 14:00, which means the total time for the round trip is 6 hours.

The total time for the round trip is the sum of the time taken for the downstream journey, the time taken for the upstream journey, and the 1-hour stop at point B.

Therefore, we can write the equation:

Time taken for downstream journey + Time taken for upstream journey + 1 hour = 6 hours

Substituting the values we calculated earlier:

(36 / (x + 3)) + (36 / (x - 3)) + 1 = 6

Now, we can solve this equation to find the value of x, which represents the speed of the boat.

Calculation

Let's solve the equation:

(36 / (x + 3)) + (36 / (x - 3)) + 1 = 6

Multiplying through by (x + 3)(x - 3) to eliminate the denominators:

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

Expanding and simplifying:

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

72x + x^2 - 117 = 6x^2 - 54

Rearranging and simplifying:

5x^2 - 72x + 63 = 0

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

Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)

where a = 5, b = -72, and c = 63.

Plugging in the values:

x = (-(-72) ± √((-72)^2 - 4 * 5 * 63)) / (2 * 5)

Simplifying:

x = (72 ± √(5184 - 1260)) / 10

x = (72 ± √3924) / 10

Calculating the square root of 3924:

√3924 ≈ 62.65

Now, we can substitute this value into the equation:

x = (72 ± 62.65) / 10

Calculating the two possible values of x:

1. x = (72 + 62.65) / 10 ≈ 13.27 2. x = (72 - 62.65) / 10 ≈ 0.73

Since the speed of the boat cannot be negative, we can conclude that the speed of the boat is approximately 13.27 km/h.

Answer

Therefore, the speed of the boat is approximately 13.27 km/h.