Question

Теплоход прошёл 72 км против течения реки и 56 км по течению, 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

Пусть х - собственная скорость теплохода, тогда (х–2) скорость против течения реки, (х+2) скорость по течению реки. Выразив время, составим уравнение:
72/(х–2)–56/(х+2)=1
(72х+144–56х+112)/(х–2)(х+2)=1
(16х+256)/(х^2–4)=1
16х+256=х^2–4
х^2–4–16х–256=0
х^2–16х–260=0
Д=/256–4•1•(-260)=/1296=36
х1=(16+36)/2=26
х2=(16–36)/2=–10 (не может являться решением)
Ответ: собственная скорость теплохода 26км/ч

Answer 2

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 more 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 and the speed of the river current is 2 km/h.

When the boat is traveling against the current, its effective speed is reduced by the speed of the current. So, the boat's speed against the current is (x - 2) km/h.

Similarly, when the boat is traveling with the current, its effective speed is increased by the speed of the current. So, the boat's speed with the current is (x + 2) km/h.

We are given that the time taken to travel against the current is 1 hour more than the time taken to travel with the current.

Using the formula distance = speed × time, we can set up the following equation:

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

Now, let's solve this equation to find the value of x.

Calculation

To solve the equation, we can cross-multiply and simplify:

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

Expanding and simplifying:

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

Rearranging and simplifying:

x^2 + 16x - 260 = 0

Now, we can solve this quadratic equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = 16, and c = -260.

Substituting these values into the quadratic formula:

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

Simplifying further:

x = (-16 ± √(256 + 1040)) / 2

x = (-16 ± √(1296)) / 2

x = (-16 ± 36) / 2

So, we have two possible solutions for x:

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.

Answer

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

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