Question

Баржа прошла по течению реки 48 км,и повернув обратно, прошла ещё 42 км,затратив на вес путь 5 часов. Найдите собственную скорость баржи,если скорость течения реки равна 5 км/ч

Answer 1

Х (км/ч) - собственная скорость баржи
х+5 (км/ч) - скорость баржи по течению реки
х-5 (км/ч) - скорость баржи против течения реки
48    (ч) - время движения баржи по течению реки
х+5
42  (ч) - время движения баржи против течения реки
х-5
Так как весь путь составил 5 часов, то составим уравнение:

48  +  42  =5
х+5    х-5

х≠5  х≠-5
Общий знаменатель: (х+5)(х-5)
48(х-5)+42(х+5)=5(х+5)(х-5)
48х-240+42х+210=5(х²-25)
90х-30=5х²-125
5х²-90х-95=0
х²-18х-19=0
Д=18²+4*19=324+76=400
х₁= 18-20 =-1  - не подходит по смыслу задачи
         2       
х₂=38 = 19 (км/ч) - собственная скорость баржи
      2
Ответ: 19 км/ч.

Answer 2

Problem Analysis

We are given that a barge traveled downstream on a river for 48 km, then turned back and traveled upstream for another 42 km. The total time taken for the round trip was 5 hours. We are also given that the speed of the river's current is 5 km/h. We need to find the speed of the barge.

Solution

Let's assume the speed of the barge is x km/h.

When the barge is traveling downstream, its effective speed is the sum of its own speed and the speed of the current. So, the effective speed downstream is (x + 5) km/h.

When the barge is traveling upstream, its effective speed is the difference between its own speed and the speed of the current. So, the effective speed upstream is (x - 5) km/h.

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

For the downstream journey: - Distance = 48 km - Speed = (x + 5) km/h - Time = Distance / Speed

For the upstream journey: - Distance = 42 km - Speed = (x - 5) km/h - Time = Distance / Speed

The total time taken for the round trip is given as 5 hours. So, the sum of the times for the downstream and upstream journeys should be equal to 5 hours.

Let's set up the equations and solve for x.

Equations

Equation 1: 48 / (x + 5) + 42 / (x - 5) = 5

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

Calculation

Using the given equation, we can solve for x:

48 / (x + 5) + 42 / (x - 5) = 5

To simplify the equation, we can multiply both sides by (x + 5)(x - 5) to eliminate the denominators:

48(x - 5) + 42(x + 5) = 5(x + 5)(x - 5)

Expanding and simplifying:

48x - 240 + 42x + 210 = 5(x^2 - 25)

90x - 30 = 5x^2 - 125

Rearranging the equation:

5x^2 - 90x + 95 = 0

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

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 5, b = -90, and c = 95.

Plugging in the values:

x = (-(-90) ± √((-90)^2 - 4 * 5 * 95)) / (2 * 5)

Simplifying:

x = (90 ± √(8100 - 1900)) / 10

x = (90 ± √6200) / 10

Calculating the square root of 6200:

√6200 ≈ 78.74

Now, we can substitute the values of x into the equation to find the two possible speeds of the barge:

x₁ = (90 + 78.74) / 10 ≈ 16.87 x₂ = (90 - 78.74) / 10 ≈ 1.13

Since the speed of the barge cannot be negative, the speed of the barge is approximately 16.87 km/h.

Answer

The speed of the barge is approximately 16.87 km/h.