Question

Расстояние между пунктами А и Б по реке 24 км. Катер проплыл от пункта А до пункта Б и вернулся обратно, затратив на весь путь 3,5 часа. Найдите собственную скорость катера(в км/ч) если скорость течения реки 2 км/ч.

Answer 1

Собственная скорость движения лодки (скорость лодки в неподвижной воде) - Х км/час

Тогда

время=путь/скорость

24/(Х+2)+24/(Х-2)=3,5

[24*(X-2)+24*(X+2)]-3,5*(X+2)*(X-2)/(X+2)*(X-2)=0

[(24*X-48+24*X+48)-3,5*(X^2-2*X+2*X-4)]/(X^2-2*X+2*X-4)=0

-3,5*X^2+48*X+14=0

Решаем квадратное уравнение с помощью дискриминанта

и получаем два корня уравнения: Х1=-0,29; Х2=14

Скорость не может быть отрицательной, поэтому Х=14 км/час

Проверяем:

24/(14+2)+24/(14-2)=3,5

24/16+24/12=3,5

1,5+2=3,5

Ответ: скорость лодки в неподвижной воде - 14 км/час

Answer 2

Calculation of the Boat's Speed

To find the boat's speed, we need to consider the speed of the current and the total time taken for the round trip.

Given: - Distance between points A and B along the river: 24 km - Speed of the river's current: 2 km/h - Total time taken for the round trip: 3.5 hours

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

To calculate the boat's speed, we can use the formula:

Speed = Distance / Time

For the downstream journey (from A to B), the boat's speed is the sum of its own speed and the speed of the current:

Boat's Speed (downstream) = x + 2 km/h

For the upstream journey (from B to A), the boat's speed is the difference between its own speed and the speed of the current:

Boat's Speed (upstream) = x - 2 km/h

Now, let's calculate the time taken for each leg of the journey.

For the downstream journey: - Distance = 24 km - Speed = Boat's Speed (downstream) = x + 2 km/h - Time = Distance / Speed = 24 / (x + 2) hours

For the upstream journey: - Distance = 24 km - Speed = Boat's Speed (upstream) = x - 2 km/h - Time = Distance / Speed = 24 / (x - 2) hours

The total time taken for the round trip is given as 3.5 hours.

Therefore, we can set up the equation:

Time (downstream) + Time (upstream) = 3.5 hours

Substituting the values:

(24 / (x + 2)) + (24 / (x - 2)) = 3.5

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

24(x - 2) + 24(x + 2) = 3.5(x + 2)(x - 2)

Simplifying the equation:

48x = 3.5(x^2 - 4)

48x = 3.5x^2 - 14

3.5x^2 - 48x - 14 = 0

Using the quadratic formula, we can solve for 'x':

x = (-b ± √(b^2 - 4ac)) / (2a)

Where: - a = 3.5 - b = -48 - c = -14

Solving this equation, we find two possible solutions for 'x': x ≈ 14.23 km/h and x ≈ -1.23 km/h. Since speed cannot be negative, we discard the negative solution.

Therefore, the boat's speed is approximately 14.23 km/h.