Question

Моторний човен пройшов 6 км за течією ріки та 3 км проти течії, витративши на весь шлях одну годину. Знайди власну швидкість човна, якщо швидкість течії ріки дорівнює 3 км/год.​

Answer 1

Пусть х км/ч - собственная скорость лодки, тогда (х + 3) км/ч - скорость лодки по течению реки, (х - 3) км/ч - скорость лодки против течения реки.  Уравнение:

6/(х+3) + 3/(х-3) = 1

6 · (х - 3) + 3 · (х + 3) = 1 · (х + 3) · (х - 3)

6х - 18 + 3х + 9 = 1 · (х² - 3²)

9х - 9 = х² - 9

х² - 9 - 9х + 9 = 0

х² - 9х = 0

D = b² - 4ac = (-9)² - 4 · 1 · 0 = 81

√D = √81 = ±9

x = (-b±√D)/2a

х₁ = (9-9)/(2·1) = 0/2 = 0 (не подходит для скорости)

х₂ = (9+9)/(2·1) = 18/2 = 9

- - - - -

Или так

х² - 9х = 0

х² = 9х

х = 9 (сократили обе части уравнения на х)

Ответ: 9 км/ч.

Проверка:

6 : (9 + 3) = 6 : 12 = 0,5 ч - время движения по течению

3 : (9 - 3) = 3 : 6 = 0,5 ч - время движения против течения

0,5 ч + 0,5 ч = 1 ч - по условию

Answer 2

Problem Analysis

We are given that a motorboat traveled 6 km downstream (with the current) and 3 km upstream (against the current) in a total of one hour. We need to find the speed of the boat in still water, given that the speed of the current is 3 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 current. Therefore, the boat's speed downstream is (x + 3) 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 current. Therefore, the boat's speed upstream is (x - 3) km/h.

We are given that the boat traveled 6 km downstream and 3 km upstream in a total of one hour. This can be represented by the equation:

6 / (x + 3) + 3 / (x - 3) = 1

To solve this equation, we can multiply both sides by (x + 3)(x - 3) to eliminate the denominators:

6(x - 3) + 3(x + 3) = (x + 3)(x - 3)

Simplifying the equation:

6x - 18 + 3x + 9 = x^2 - 9

Combining like terms:

9x - 9 = x^2 - 9

Rearranging the equation:

x^2 - 9x = 0

Factoring out an x:

x(x - 9) = 0

Setting each factor equal to zero:

x = 0 or x - 9 = 0

Therefore, the possible values for x are x = 0 or x = 9.

However, the speed of the boat cannot be zero, so the only valid solution is x = 9.

Answer

The speed of the boat in still water is 9 km/h.