Question

Баржа прошла по течению реки 40км и,повернув обратно,прошла еще 30км,затратив на весь путь 5часов.Найдите собственную скорость баржи,если скорость течения реки равна 5км/ч.Помогите пожалуйста

Answer 1

                       S             V                     t
по теч .        40км      х + 5км/ч         40/(х +5) ч
пр. теч.       30 км      х - 5 км/ч       30/(х -5) ч
V собств. = х км/ч
Vтеч. = 5 км/ч
Составим уравнение:
40/(х + 5) +  30/(х -5) = 5 | * (x +5)(x - 5)≠ 0
                                               x≠ -5,   x≠ 5
40(x - 5) +30(x+5) = 5(x² -25)
40x -200 +30x +150 = 5x² -125,
5x²  -70x -75 = 0
x² - 14x -  15 = 0
По т. Виета: х1 = -1 ( не подходит по условию задачи)
                      х2 = 15 (км/ч) - V собств.
Ответ: Vсоств. = 15 км/ч 

Answer 2

Problem Analysis

We are given that a barge traveled downstream for 40 km, then turned back and traveled upstream for 30 km. The total time taken for the entire journey was 5 hours. We are also given that the speed of the river 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, it gets a boost from the river current, so its effective speed is the sum of its own speed and the speed of the current. Therefore, the effective speed downstream is (x + 5) km/h.

When the barge is traveling upstream, it has to overcome the resistance of the river current, so its effective speed is the difference between its own speed and the speed of the current. Therefore, 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.

The time taken to travel downstream for 40 km is given by: 40 km = (x + 5) km/h × t1

The time taken to travel upstream for 30 km is given by: 30 km = (x - 5) km/h × t2

We also know that the total time taken for the entire journey is 5 hours: t1 + t2 = 5 hours

We can solve this system of equations to find the value of x, the speed of the barge.

Calculation

Let's solve the system of equations:

From the first equation, we can isolate t1: t1 = 40 km / (x + 5) km/h

From the second equation, we can isolate t2: t2 = 30 km / (x - 5) km/h

Substituting these values into the third equation, we get: 40 km / (x + 5) km/h + 30 km / (x - 5) km/h = 5 hours

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

40 km(x - 5) + 30 km(x + 5) = 5 hours(x + 5)(x - 5)

Expanding and simplifying the equation:

40x - 200 + 30x + 150 = 5(x^2 - 25)

70x - 50 = 5x^2 - 125

Rearranging the equation:

5x^2 - 70x + 75 = 0

We can solve this quadratic equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

where a = 5, b = -70, and c = 75.

Calculating the values of x using the quadratic formula:

x = (-(-70) ± √((-70)^2 - 4 * 5 * 75)) / (2 * 5)

x = (70 ± √(4900 - 1500)) / 10

x = (70 ± √3400) / 10

x = (70 ± 58.31) / 10

So, the possible values for x are:

x1 = (70 + 58.31) / 10 ≈ 12.83 km/h

x2 = (70 - 58.31) / 10 ≈ 1.17 km/h

Since the speed of the barge cannot be negative, the speed of the barge is approximately 12.83 km/h.

Answer

The speed of the barge is approximately 12.83 km/h.