Question

Катер прошел 12 км против течения реки и 5 км по течению.При этом он затратил столько времени,сколько ему потребовалось бы,если бы он шел 18 км по озеру.Какова собственная скорость катера,если известно,что скорость течения реки равна 3 км/ч?

Answer 1

Problem Analysis

We are given the following information: - The boat traveled 12 km against the current of a river and 5 km with the current. - The time taken for this journey is the same as it would have taken to travel 18 km on a lake. - The speed of the river current is 3 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 against the current, its effective speed is reduced by the speed of the current. So, the effective speed is (x - 3) km/h.

When the boat is traveling with the current, its effective speed is increased by the speed of the current. So, the effective speed is (x + 3) km/h.

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

For the journey against the current: - Distance = 12 km - Speed = (x - 3) km/h - Time = Distance / Speed = 12 / (x - 3) hours

For the journey with the current: - Distance = 5 km - Speed = (x + 3) km/h - Time = Distance / Speed = 5 / (x + 3) hours

According to the given information, the time taken for both journeys is the same as it would have taken to travel 18 km on a lake. So, we can set up the following equation:

12 / (x - 3) = 18 / x

Let's solve this equation to find the value of x.

Calculation

To solve the equation, we can cross-multiply:

12x = 18(x - 3)

Expanding the equation:

12x = 18x - 54

Rearranging the terms:

6x = 54

Dividing both sides by 6:

x = 9

Answer

The speed of the boat is 9 km/h.

Verification

To verify our answer, let's check if the time taken for both parts of the journey is the same.

For the journey against the current: - Distance = 12 km - Speed = (9 - 3) km/h = 6 km/h - Time = Distance / Speed = 12 / 6 = 2 hours

For the journey with the current: - Distance = 5 km - Speed = (9 + 3) km/h = 12 km/h - Time = Distance / Speed = 5 / 12 ≈ 0.4167 hours

The total time taken for both parts of the journey is approximately 2.4167 hours.

Now, let's calculate the time it would have taken to travel 18 km on a lake: - Distance = 18 km - Speed = 9 km/h - Time = Distance / Speed = 18 / 9 = 2 hours

As we can see, the time taken for both parts of the journey is the same as it would have taken to travel 18 km on a lake. Therefore, our answer is verified.

Conclusion

The speed of the boat is 9 km/h.