Question

1.     Моторная лодка прошла 45 км по течению реки и столько же против течения затратив всего 14 часов. Найдите скорость лодки в стоячей воде, если скорость течения реки равна 2 км/ч. ​

Answer 1

Пусть собственная скорость лодки х км/ч, тогда скорость лодки по течению реки (х + 2) км/ч, а скорость лодки против течения реки (х - 2) км/ч. Расстояние в 45 километров лодка прошла по течению реки за 45/(х + 2) часа, а против течения реки за 45/(х - 2) часа. По условию задачи известно, что на весь путь лодка затратила (45/(х + 2) + 45/(х - 2)) часа или 14 часов. Составим уравнение и решим его.
45/(х + 2) + 45/(х - 2) = 14;
45(х - 2) + 45(х + 2) = 14(х^2 - 4);
45х - 90 + 45х + 90 = 14х^2 - 56;
14х^2 - 90х - 56 = 0;
7х^2 - 45х - 28 = 0;
D = 45^2 - 4 * 7 * (-28) = 2809; √D = 53;
х1 = (45 + 53)/14 = 7 (км/ч);
х2 = (45 - 53)/14 = -8/14 - скорость не может быть отрицательной.
Ответ. 7 км/ч.

не спалитесь

Answer 2

Problem Analysis

We are given that a motorboat traveled 45 km downstream (with the current) and the same distance upstream (against the current), taking a total of 14 hours. We need to find the speed of the boat in still water, given that the speed of the river current is 2 km/h.

Solution

Let's assume the speed of the boat in still water is x km/h. Since the boat is traveling both downstream and upstream, we can use the formula:

Distance = Speed × Time

For the downstream journey, the boat's effective speed is the sum of its speed in still water and the speed of the current, which is (x + 2) km/h. The distance traveled downstream is 45 km. Therefore, the time taken for the downstream journey is:

45 km = (x + 2) km/h × Time

For the upstream journey, the boat's effective speed is the difference between its speed in still water and the speed of the current, which is (x - 2) km/h. The distance traveled upstream is also 45 km. Therefore, the time taken for the upstream journey is:

45 km = (x - 2) km/h × Time

Since the total time taken for both journeys is 14 hours, we can write the equation:

Time downstream + Time upstream = 14 hours

Substituting the expressions for the times from the previous equations, we get:

(45 km) / (x + 2) km/h + (45 km) / (x - 2) km/h = 14 hours

To solve this equation, we can multiply both sides by (x + 2)(x - 2) to eliminate the denominators:

45(x - 2) + 45(x + 2) = 14(x + 2)(x - 2)

Simplifying the equation:

45x - 90 + 45x + 90 = 14(x^2 - 4)

90x = 14x^2 - 56

14x^2 - 90x - 56 = 0

Now we can solve this quadratic equation for x using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

where a = 14, b = -90, and c = -56.

Calculating the values:

x = (-(-90) ± √((-90)^2 - 4 * 14 * (-56))) / (2 * 14)

x = (90 ± √(8100 + 3136)) / 28

x = (90 ± √11236) / 28

x = (90 ± 106) / 28

So, we have two possible solutions for x:

x = (90 + 106) / 28 = 196 / 28 = 7 km/h

x = (90 - 106) / 28 = -16 / 28 = -0.57 km/h

Since the speed of the boat cannot be negative, we can conclude that the speed of the boat in still water is 7 km/h.

Answer

The speed of the motorboat in still water is 7 km/h.