Question

Катер прошел 80 км по течению реки и вернулся обратно затратив на весь путь 9 часов. Найдите собственную скорость катера, если скорость течения реки 18 км/ч.

Answer 1

Собственная скорость катера = х(км/ч)
Скорость по течению = (х + 18) км/ч
Скорость против течения = (х - 18)км/ч
Время по течению = 80/(х + 18) ч
Время против течения = 80(х - 18) ч
Составим уравнение:
80/(х + 18) + 80/(х - 18) = 9
80(х-18) + 80(х + 18) = 9(х^2 - 324)
80x - 1440 + 80x + 1440 = 9x^2 - 2916
- 9x^2 + 160x + 2916 = 0
  9x^2 - 160x -2916 = 0
D = 25600-4*-2916*9 = 25600+104976 = 130576;   √D ≈ 361,4
x1 = (160 + 361,4)/18 ≈ 29
x2 = - 11 (не подходит по условию задачи)
Ответ: 29км\ч - собственная скорость катера. 

Answer 2

Problem Analysis

We are given that a boat traveled 80 km upstream and then returned back, spending a total of 9 hours for the entire trip. The speed of the river's current is given as 18 km/h. We need to find the speed of the boat.

Solution

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

When the boat is traveling upstream, its effective speed is reduced by the speed of the river's current. So, the boat's speed relative to the ground is (x - 18) km/h.

When the boat is traveling downstream, its effective speed is increased by the speed of the river's current. So, the boat's speed relative to the ground is (x + 18) km/h.

We can use the formula distance = speed × time to calculate the time taken for each leg of the journey.

1. Upstream journey: - Distance = 80 km - Speed = (x - 18) km/h - Time = Distance / Speed = 80 / (x - 18) hours

2. Downstream journey: - Distance = 80 km - Speed = (x + 18) km/h - Time = Distance / Speed = 80 / (x + 18) hours

According to the problem, the total time taken for the entire trip is 9 hours. So, we can write the equation:

Time taken for upstream journey + Time taken for downstream journey = 9 hours

Substituting the values:

80 / (x - 18) + 80 / (x + 18) = 9

To solve this equation, we can multiply both sides by (x - 18)(x + 18) to eliminate the denominators:

80(x + 18) + 80(x - 18) = 9(x - 18)(x + 18)

Simplifying the equation:

80x + 1440 + 80x - 1440 = 9(x^2 - 18^2)

160x = 9(x^2 - 324)

160x = 9x^2 - 2916

Rearranging the equation:

9x^2 - 160x - 2916 = 0

We can solve this quadratic equation to find the value of x.