Question

Лодка вышла из пункта А в 8.00,и,пройдя вниз по течению реки 24 км прибыла в пункт В.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 равно 2 км/ч

Answer 1

Лодка прошла 24 км по течению реки скоростью: v+2, затратив на это 24/(v+2) часов.
Затем лодка прошла 24 км против течения реки скоростью: v-2, затратив на это 24/(v-2) часов.
Всего в пути лодка находилась: 14-8-1=5 часов.
Составляем уравнение: 24/(v+2) + 24/(v-2) = 5
24(v-2+v+2)/((v+2)(v-2)) = 5
48v = 5(v²-4)  (v ≠ +-2)
5v² - 48v - 20 = 0
D = 48² + 4·5·20 = 2704 = 52²
v = (48 +- 52)/10={-0,4; 10}.
v ≠ +-2 км/ч. Также, по условию задачи, скорость лодки должна превышать скорость течения, поэтому: v > 2 км/ч.
Учитывая это условие, получаем: v = 10 км/ч

Answer 2

Problem Analysis

We are given the following information: - A boat leaves point A at 8:00 and travels downstream for 24 km to reach point B. - The boat stops at point B for 1 hour. - The boat then returns upstream to point A and arrives at 14:00 on the same day. - The speed of the river current is 2 km/h.

We need to determine the speed of the boat.

Solution

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

When the boat is traveling downstream, it is aided by the current, so 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, it is opposed by the current, so 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 use the formula distance = speed × time to calculate the distances traveled.

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

1. Downstream journey from A to B: - Distance = 24 km - Speed = (x + 2) km/h - Time = distance / speed = 24 / (x + 2) hours

2. Upstream journey from B to A: - Distance = 24 km - Speed = (x - 2) km/h - Time = distance / speed = 24 / (x - 2) hours

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

Total time = (24 / (x + 2)) + (24 / (x - 2)) + 1

We know that the boat leaves point A at 8:00 and arrives back at point A at 14:00, which is a total of 6 hours. Therefore, we can set up the following equation:

(24 / (x + 2)) + (24 / (x - 2)) + 1 = 6

Now, let's solve this equation to find the value of x, which represents the speed of the boat.

Calculation

To solve the equation, we can multiply through by the common denominator (x + 2)(x - 2) to eliminate the fractions:

24(x - 2) + 24(x + 2) + (x + 2)(x - 2) = 6(x + 2)(x - 2)

Simplifying the equation:

24x - 48 + 24x + 48 + x^2 - 4 = 6(x^2 - 4)

48x + x^2 - 4 = 6x^2 - 24

Rearranging the equation:

5x^2 - 48x - 20 = 0

Using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

where a = 5, b = -48, and c = -20.

Plugging in the values:

x = (-(-48) ± √((-48)^2 - 4 * 5 * -20)) / (2 * 5)

Simplifying:

x = (48 ± √(2304 + 400)) / 10

x = (48 ± √(2704)) / 10

x = (48 ± 52) / 10

x = (48 + 52) / 10 or x = (48 - 52) / 10

x = 100 / 10 or x = -4 / 10

x = 10 or x = -0.4

Since the speed of the boat cannot be negative, the speed of the boat is 10 km/h.

Answer

The speed of the boat is 10 km/h.

Explanation

The boat travels downstream at a speed of 10 km/h (its own speed) + 2 km/h (the speed of the current), which gives an effective speed of 12 km/h. It takes 2 hours to cover the distance of 24 km downstream.

On the return journey upstream, the boat travels at a speed of 10 km/h (its own speed) - 2 km/h (the speed of the current), which gives an effective speed of 8 km/h. It also takes 2 hours to cover the distance of 24 km upstream.

The total time for the round trip is 2 hours downstream + 1 hour stop at point B + 2 hours upstream = 5 hours. This matches the given information that the boat arrives back at point A at 14:00, which is 6 hours after it left at 8:00.

Therefore, the speed of the boat is indeed 10 km/h, and it covers the distance of 24 km downstream and 24 km upstream in a total of 5 hours.