Question

Теплоход прошел 72 км против течения реки и 56 км по течению реки, затратив на путь против течения на 1 ч больше,чем на путь по течению. Найдите собственную скорость теплохода,если скорость течения реки составляет 2 км\ ч

Answer 1

Собственная скорость теплохода   -  х км/ч.

Против течения:
Расстояние  72  км
Скорость  (х-2) км/ч
Время в пути   72/(х-2)   ч.

По течению:
Расстояние  56 км
Скорость  (х+2)  км/ч
Время в пути    56/(х+2) км/ч

Время , затраченное на путь против течения на  1 час больше.
Уравнение.
72/ (х-2)    -   56/(х+2)  =  1             | × (x-2)(x+2)
72(x+2)  - 56(x-2)= 1 (x-2)(x+2)
72x+144  - 56x + 112 = x²-4
16x + 256 = x²-4
x²-4-16x-256=0
 x²-16x-260=0
D= 16²- 4*1*(-260) = 256+1040=1296 ; √D= 36
D>0  - два корня уравнения
х₁= (16-36)/2 = -20/2 =-10   - не удовл. условию задачи
х₂ = (16+36)/2 = 52/2=26 (км/ч)  собственная скорость теплохода.

Ответ: 26 км/ч.   

Answer 2

Пусть х км/ч собственная скорость теплохода
х+2 - скорость по течению реки,
х-2 - скорость против течения реки
Зная, что против течения теплоход прошёл расстояние 72км - 
72 : (х-2) - это время против течения и 
56 : (х+2) - это время по течению.
Зная, что разница во времени сост 1 ч, сост ур-ие:
72/(х-2) - 56/(х+2)= 1
72х+144 - 56х+112 = (х-2) (х+2)
16х+256 = х²-4
-х²+16х+256+4 = 0
-х²+16х+260 = 0
Д=в²-4ас
Д= 256 - 4 (-1)* (260)
Д = 1296
х₁ = -в+√Д / 2а         х₂ = -в-√Д / 2а
х= -16+36 / -2            х= -16-36 / -2
х= -10                        х= 26 
Скорость теплохода 26 км/ч

Answer 3

Problem Analysis

We are given that a boat traveled 72 km against the current of a river and 56 km with the current. The time taken to travel against the current was 1 hour longer than the time taken to travel with the current. We need to find the speed of the boat.

Solution

Let's assume the speed of the boat in still water is x km/h. The speed of the current is given as 2 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 - 2) 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 + 2) km/h.

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

For the journey against the current: - Distance = 72 km - Speed = (x - 2) km/h - Time = 72 / (x - 2) hours

For the journey with the current: - Distance = 56 km - Speed = (x + 2) km/h - Time = 56 / (x + 2) hours

According to the problem, the time taken to travel against the current is 1 hour longer than the time taken to travel with the current. So, we can set up the following equation:

72 / (x - 2) = 56 / (x + 2) + 1

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

Calculation

To solve the equation, we can start by simplifying it:

72 / (x - 2) = 56 / (x + 2) + 1

Multiplying both sides of the equation by (x - 2) and (x + 2) to eliminate the denominators:

72(x + 2) = 56(x - 2) + (x - 2)(x + 2)

Expanding and simplifying:

72x + 144 = 56x - 112 + x^2 - 4

Rearranging the equation:

x^2 + 16x - 260 = 0

Now we have a quadratic equation. We can solve it by factoring or using the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = 16, and c = -260. Substituting these values into the formula:

x = (-16 ± √(16^2 - 4(1)(-260))) / (2(1))

Simplifying:

x = (-16 ± √(256 + 1040)) / 2

x = (-16 ± √1296) / 2

x = (-16 ± 36) / 2

We have two possible solutions:

x1 = (-16 + 36) / 2 = 20 / 2 = 10 x2 = (-16 - 36) / 2 = -52 / 2 = -26

Since the speed of the boat cannot be negative, we discard the negative solution. Therefore, the speed of the boat in still water is 10 km/h.

Answer

The speed of the boat in still water is 10 km/h.