Question

Катер прошел 12 км по течению реки и 4 км против течения реки, затратив на весь путь 2 ч. Чему равна собственная скорость катера, если скорость течения реки равна 4 км/ч?

Answer 1

Пусть х км/ч скорость катера. Тогда скорость по течению реки (х+4) км/ч, а против течения - (х-4). Составим уравнение:

(12/(х+4))+(4/(х-4))=2

6(х-4)+2(х+4)=(х-4)(х+4)

6х-24+2х+8=х²-4

х²-8х+12=0

х₁=6, х₂=2 не удовлет. условию задачи

Ответ: 6 км/ч

Answer 2

Problem Analysis

We are given that a boat traveled 12 km downstream and 4 km upstream in a river. The total time taken for the entire journey was 2 hours. We need to find the speed of the boat in still water, given that the speed of the river current is 4 km/h.

Solution

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

When the boat is traveling downstream, its effective speed is the sum of its own speed and the speed of the river current. So, the effective speed is (x + 4) km/h.

When the boat is traveling upstream, its effective speed is the difference between its own speed and the speed of the river current. So, the effective speed is (x - 4) km/h.

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

1. Downstream journey: - Distance = 12 km - Speed = (x + 4) km/h - Time = Distance / Speed = 12 / (x + 4) hours

2. Upstream journey: - Distance = 4 km - Speed = (x - 4) km/h - Time = Distance / Speed = 4 / (x - 4) hours

The total time taken for the entire journey is given as 2 hours. So, we can write the equation:

Time taken downstream + Time taken upstream = 2

(12 / (x + 4)) + (4 / (x - 4)) = 2

To solve this equation, we can multiply both sides by (x + 4)(x - 4) to eliminate the denominators:

12(x - 4) + 4(x + 4) = 2(x + 4)(x - 4)

Simplifying the equation:

12x - 48 + 4x + 16 = 2(x^2 - 16)

16x - 32 = 2x^2 - 32

Rearranging the equation:

2x^2 - 16x = 0

Factoring out 2x:

2x(x - 8) = 0

Setting each factor equal to zero:

2x = 0 or x - 8 = 0

Solving for x:

x = 0 or x = 8

Since the speed of the boat cannot be zero, the only valid solution is:

x = 8 km/h

Therefore, the speed of the boat in still water is 8 km/h.

Answer

The speed of the boat in still water is 8 km/h.