Question

Расстояние между пристанями А и Б равно 48 км.Из А в Б по течению реки отправился плот, а через час за ним отправилась моторная лодка, которая, прибыв в пункт Б , тотчас повернула обратно и возвратилась в А.К этому времени плот прошел 25 км. Найдите скорость лодки в неподвижной воде, если скорость течения реки равна 5 км/ч

Answer 1

Скорость плота = скорости течения реки = 5 км/ч
1) 25 : 5 = 5 часов - время, за которое проплыл плот 25 км

2) 5 - 1 = 4 часа - время, за которое лодка проплыла из пункта А в пункт В и обратно

3) х км/ч - скорость лодки в неподвижной воде

48/(х + 5) + 48/(х - 5) = 4
48х - 240 + 48х + 240 - 4х² + 100 = 0
96х - 4х² + 100 = 0
х² - 24х + 25 = 0
По теореме Виета х₁ = -1   х₂ = 25
х₁ - не удовлетворяет условию задачи, т.к. скорость отрицательной быть не может

25 км/ч - скорость лодки в неподвижной воде

Answer 2

Problem Analysis

We are given that the distance between two docks, A and B, is 48 km. A raft starts from dock A and after an hour, a motorboat starts from dock A and travels towards dock B. When the motorboat reaches dock B, it immediately turns back and returns to dock A. By the time the motorboat reaches dock B, the raft has traveled 25 km. We need to find the speed of the motorboat in still water, given that the speed of the river current is 5 km/h.

Solution

Let's assume the speed of the motorboat in still water is x km/h. The speed of the river current is given as 5 km/h.

When the motorboat is traveling downstream (from A to B), it gets an additional speed boost from the river current. So, the effective speed of the motorboat downstream is (x + 5) km/h.

When the motorboat is traveling upstream (from B to A), it has to overcome the speed of the river current. So, the effective speed of the motorboat upstream is (x - 5) km/h.

We know that the raft has traveled 25 km by the time the motorboat reaches dock B. This means that the motorboat has traveled the remaining distance of (48 - 25) = 23 km.

To find the time taken by the motorboat to travel from A to B, we can use the formula:

Time = Distance / Speed

The time taken by the motorboat to travel downstream is given by:

Time downstream = Distance downstream / Speed downstream

Substituting the values, we get:

Time downstream = 25 km / (x + 5) km/h

The time taken by the motorboat to travel upstream is given by:

Time upstream = Distance upstream / Speed upstream

Substituting the values, we get:

Time upstream = 23 km / (x - 5) km/h

Since the motorboat starts an hour after the raft, the total time taken by the motorboat for the round trip is 1 hour more than the time taken by the raft. Therefore, we can write:

Time downstream + Time upstream = Time raft + 1 hour

Substituting the values, we get:

25 / (x + 5) + 23 / (x - 5) = 1

Now, we can solve this equation to find the value of x, which represents the speed of the motorboat in still water.

Calculation

Let's solve the equation:

25 / (x + 5) + 23 / (x - 5) = 1

Multiplying through by (x + 5)(x - 5) to eliminate the denominators, we get:

25(x - 5) + 23(x + 5) = (x + 5)(x - 5)

Expanding and simplifying, we get:

25x - 125 + 23x + 115 = x^2 - 25

Combining like terms, we get:

48x - 10 = x^2 - 25

Rearranging the equation, we get:

x^2 - 48x + 15 = 0

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

Using the quadratic formula, we have:

x = (-b ± √(b^2 - 4ac)) / (2a)

Substituting the values, we get:

x = (48 ± √(48^2 - 4 * 1 * 15)) / (2 * 1)

Simplifying, we get:

x = (48 ± √(2304 - 60)) / 2

x = (48 ± √(2244)) / 2

x = (48 ± 47.38) / 2

Simplifying further, we get two possible values for x:

x = (48 + 47.38) / 2 ≈ 47.69

x = (48 - 47.38) / 2 ≈ 0.31

Since the speed of the motorboat cannot be negative, we can discard the second solution.

Therefore, the speed of the motorboat in still water is approximately 47.69 km/h.

Answer

The speed of the motorboat in still water is approximately 47.69 km/h.