Question

Моторная лодка прошла 36 км по течению реки и вернулась обратно, потратив на весь путь 5 часов. Скорость течения реки 3 км/ ч. Найдите скорость лодки в неподвижной воде.

Answer 1

Пусть х км/ч - собственная скорость лодки
(х+3) км/ч - скорость лодки по течению
(х-3) км/ч - скорость лодки против течения
t=S:V
36/(x+3) + 36/(х-3)=5
36х-108+36+108+5х²-45
5х²-72х-45=0
D=5184+900=6084=78²
х=(72+78)/10=15(км/ч) -  собственная скорость лодки

Answer 2

Problem Analysis

We are given that a motorboat traveled 36 km upstream in a river and then returned back, taking a total of 5 hours for the entire trip. The speed of the river's current is given as 3 km/h. We need to find the speed of the motorboat in still water.

Solution

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

When the motorboat is traveling upstream against the current, its effective speed is reduced by the speed of the current. So, the speed of the motorboat relative to the ground is (x - 3) km/h.

When the motorboat is traveling downstream with the current, its effective speed is increased by the speed of the current. So, the speed of the motorboat relative to the ground is (x + 3) km/h.

We are given that the motorboat traveled a total distance of 36 km upstream and then returned back, taking a total of 5 hours for the entire trip.

Let's calculate the time taken for the upstream journey and the downstream journey separately.

Upstream Journey

The distance traveled upstream is 36 km, and the speed of the motorboat relative to the ground is (x - 3) km/h.

Using the formula time = distance / speed, we can calculate the time taken for the upstream journey as: time_upstream = 36 / (x - 3) hours.

Downstream Journey

The distance traveled downstream is also 36 km, and the speed of the motorboat relative to the ground is (x + 3) km/h.

Using the formula time = distance / speed, we can calculate the time taken for the downstream journey as: time_downstream = 36 / (x + 3) hours.

Total Time

The total time taken for the entire trip is given as 5 hours.

Since the motorboat traveled upstream and then returned downstream, the total time can be expressed as the sum of the time taken for the upstream journey and the time taken for the downstream journey: time_upstream + time_downstream = 5 hours.

Now, we can substitute the values of time_upstream and time_downstream into the equation above and solve for x.

Calculation

Let's substitute the values and solve for x:

36 / (x - 3) + 36 / (x + 3) = 5

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

36(x + 3) + 36(x - 3) = 5(x - 3)(x + 3)

Simplifying the equation:

36x + 108 + 36x - 108 = 5(x^2 - 9)

72x = 5x^2 - 45

Rearranging the equation:

5x^2 - 72x - 45 = 0

Now, we can solve this quadratic equation to find the value of x.

Quadratic Equation Solution

Using the quadratic formula x = (-b ± √(b^2 - 4ac)) / (2a), where a = 5, b = -72, and c = -45, we can find the values of x.

x = (-(-72) ± √((-72)^2 - 4 * 5 * (-45))) / (2 * 5)

Simplifying the equation:

x = (72 ± √(5184 + 900)) / 10

x = (72 ± √(6084)) / 10

x = (72 ± 78) / 10

So, we have two possible values for x:

1. x = (72 + 78) / 10 = 15 km/h 2. x = (72 - 78) / 10 = -0.6 km/h

Since the speed of the motorboat cannot be negative, we can discard the second solution.

Answer

Therefore, the speed of the motorboat in still water is 15 km/h.