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

12/(х+4) + 4/(х-4) = 2
12(х-4) + 4(х+4) = 2(х-4)(х+4)
6х - 24 + 2х + 8 = х²-16х²-16 - 8х + 16 = 0
х²- 8х = 0
х(х-8) = 0
х = 8

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 speed in still water and the speed of the river current. So, the effective speed downstream is (x + 4) km/h.

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

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

The time taken to travel downstream for 12 km is given by 12 / (x + 4) hours.

The time taken to travel upstream for 4 km is given by 4 / (x - 4) hours.

Since the total time taken for the entire journey is 2 hours, we can set up the equation:

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

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

Calculation

Let's solve the equation:

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

To simplify the 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 further:

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.

Answer

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