Question

1)Катер проплыл 72 км между пристанями по течению реки за 2 часа, а против течения-за 3 ч. За сколько часов это расстояние проплывут плоты? 2)Лодка плывёт по течению реки. Гребец уронил в воду шляпу и, не заметив этого, продолжал плыть дальше. Какое расстояние будет между лодкой и шляпой через 10мин, если собственная скорость лодки-9км/ч? 3)Катер проплыл 72км за 3 ч, а против течения-за 4ч. Найдите скорость течения реки.

Answer 1

Для решения задачи надо понять, что
ПО РЕКЕ  плот плывет со скоростью РЕКИ - Vр
ПО ТЕЧЕНИЮ - скорости катера и реки СУММИРУЮТСЯ = Vк+V р
ПРОТИВ ТЕЧЕНИЯ - скорости ВЫЧИТАЮТСЯ  = Vк-Vр
И главную формулу -  ПУТЬ = СКОРОСТЬ*ВРЕМЯ
Вопрос 1
Пишем такие уравнения.
1) (Vк+Vр)*2=72 км
2)  (Vк-Vр)*3 = 72 км
 Найти Vр=? - скорость реки     
Приравниваем 1) и 2)
3 )2Vк +2Vр  =3Vк - 3Vр
Приводим общие члены
(3+2)Vр =5Vр = (3-2)Vк = Vк или  
4)Vк = 5Vр  подставляем в любое, например, в 1)
5) 12Vр  =72    Vр = 72/12 = 6 км/час (Vк = 30 км/час)
6) Время ПЛОТА - со скоростью реки
Тпл = 72 км / 6 км/час =12 час
Ответ: Плот пройдет расстояние за 12 час.
Вопрос 2
Дано - скорость шляпы = скорости реки = 6 км/час.
Скорость лодки Vл = 9 км/час.
Найти: Разность путей через Т=10 мин.
1) Разность скоростей лодки и шляпы
Vл - Vр = 9-6 =3 км/час.
2) Расстояние через 10 мин
S = (Vл- Vр)*t =3 км/час * (10/60) час = 0,5 км = 500 м
Ответ: Расстояние будет 500 м
Вопрос 3
Решение
(Vк+Vр)*3=72=(Vк-Vр)*4
3Vк+3Vр = 4Vк -4Vр
Vк =7Vр
24Vр = 72
V  =3
Ответ: скорость реки 3 км/час

Answer 2

1) Катер проплыл 72 км между пристанями по течению реки за 2 часа, а против течения - за 3 ч. За сколько часов это расстояние проплывут плоты?

To find out how long it will take for the rafts to cover the same distance, we need to determine the speed of the current. We can do this by finding the difference between the speed of the boat when going with the current and against the current.

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

When the boat is going with the current, its effective speed is the sum of the speed of the boat and the speed of the current: (x + y) km/h. According to the given information, the boat covers a distance of 72 km in 2 hours, so we can set up the equation:

72 km = (x + y) km/h * 2 h

When the boat is going against the current, its effective speed is the difference between the speed of the boat and the speed of the current: (x - y) km/h. According to the given information, the boat covers the same distance of 72 km in 3 hours, so we can set up the equation:

72 km = (x - y) km/h * 3 h

We now have a system of two equations with two unknowns. We can solve this system of equations to find the values of x and y.

Let's solve the system of equations:

Equation 1: 72 km = (x + y) km/h * 2 h

Equation 2: 72 km = (x - y) km/h * 3 h

To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of substitution:

From Equation 1, we can isolate x in terms of y:

x = 72 km / (2 h) - y km/h

Simplifying, we get:

x = 36 km/h - y km/h

Substituting this value of x into Equation 2, we get:

72 km = (36 km/h - y km/h - y km/h) * 3 h

Simplifying, we get:

72 km = (36 km/h - 2y km/h) * 3 h

Expanding, we get:

72 km = 108 km/h - 6y km/h

Rearranging the equation, we get:

6y km/h = 108 km/h - 72 km

Simplifying, we get:

6y km/h = 36 km

Dividing both sides by 6 km/h, we get:

y = 6 km/h

Now that we have found the speed of the current, we can calculate the time it will take for the rafts to cover the same distance.

The speed of the rafts will be the speed of the boat minus the speed of the current: x - y = x - 6 km/h.

To find the time it will take for the rafts to cover the distance of 72 km, we can use the formula:

Time = Distance / Speed

Substituting the values, we get:

Time = 72 km / (x - 6 km/h)

Let's calculate the time:

Time = 72 km / (x - 6 km/h)

Since we don't have the value of x, we cannot calculate the exact time it will take for the rafts to cover the distance. However, we can provide the formula to calculate it once the value of x is known.

2) Лодка плывёт по течению реки. Гребец уронил в воду шляпу и, не заметив этого, продолжал плыть дальше. Какое расстояние будет между лодкой и шляпой через 10 мин, если собственная скорость лодки - 9 км/ч?

To find the distance between the boat and the hat after 10 minutes, we need to calculate the distance the boat travels in that time.

Given that the boat is moving with the current, its effective speed is the sum of its own speed and the speed of the current. Let's assume the speed of the current is c km/h.

The boat's speed is given as 9 km/h, so its effective speed is 9 km/h + c km/h.

To calculate the distance the boat travels in 10 minutes (0.167 hours), we can use the formula:

Distance = Speed × Time

Substituting the values, we get:

Distance = (9 km/h + c km/h) × 0.167 h

Simplifying, we get:

Distance = (9 + c) × 0.167 km

Now we have the distance the boat travels in 10 minutes. However, we need to find the distance between the boat and the hat.

Since the boat is moving with the current, the hat will also be carried by the current. Therefore, the distance between the boat and the hat will remain constant.

Hence, the distance between the boat and the hat after 10 minutes will be the same as the distance between them initially.

3) Катер проплыл 72 км за 3 ч, а против течения - за 4 ч. Найдите скорость течения реки.

To find the speed of the current, we can use the given information about the boat's speed and the time it takes to cover a certain distance.

Let's assume the speed of the boat in still water is b km/h and the speed of the current is c km/h.

When the boat is going with the current, its effective speed is the sum of the speed of the boat and the speed of the current: (b + c) km/h. According to the given information, the boat covers a distance of 72 km in 3 hours, so we can set up the equation:

72 km = (b + c) km/h * 3 h

When the boat is going against the current, its effective speed is the difference between the speed of the boat and the speed of the current: (b - c) km/h. According to the given information, the boat covers the same distance of 72 km in 4 hours, so we can set up the equation:

72 km = (b - c) km/h * 4 h

We now have a system of two equations with two unknowns. We can solve this system of equations to find the values of b and c.

Let's solve the system of equations:

Equation 1: 72 km = (b + c) km/h * 3 h

Equation 2: 72 km = (b - c) km/h * 4 h

To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of substitution:

From Equation 1, we can isolate b in terms of c:

b = 72 km / (3 h) - c km/h

Simplifying, we get:

b = 24 km/h - c km/h

Substituting this value of b into Equation 2, we get:

72 km = (24 km/h - c km/h - c km/h) * 4 h

Simplifying, we get:

72 km = (24 km/h - 2c km/h) * 4 h

Expanding, we get:

72 km = 96 km/h - 8c km/h

Rearranging the equation, we get:

8c km/h = 96 km/h - 72 km

Simplifying, we get:

8c km/h = 24 km

Dividing both sides by 8 km/h, we get:

c = 3 km/h

Therefore, the speed of the current is 3 km/h.