Question

1)Моторная лодка прошла 10км по течению реки и 12 км против течения,затратив на весь путь 2 ч. Скорость течения реки равна 3 км\ч. Найдите скорость реки

Answer 1

Воот. Это мы в классе решали) Это правильно решено)

Answer 2

Problem Analysis

We are given that a motorboat traveled 10 km downstream (with the current) and 12 km upstream (against the current) in a total time of 2 hours. The speed of the river's current is given as 3 km/h. We need to find the speed of the river.

Solution

Let's assume the speed of the motorboat in still water is x km/h.

When the motorboat is traveling downstream, its effective speed is the sum of its speed in still water and the speed of the current. Therefore, the speed downstream is (x + 3) km/h.

When the motorboat is traveling upstream, its effective speed is the difference between its speed in still water and the speed of the current. Therefore, the speed upstream is (x - 3) 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 is given by: 10 km = (x + 3) km/h × t1 hours (Equation 1)

The time taken to travel upstream is given by: 12 km = (x - 3) km/h × t2 hours (Equation 2)

We are also given that the total time for the journey is 2 hours: t1 + t2 = 2 hours (Equation 3)

We can solve this system of equations to find the value of x, which represents the speed of the river.

Solving the Equations

Let's solve the system of equations to find the value of x.

From Equation 1, we can express t1 in terms of x: t1 = 10 km / (x + 3) km/h

From Equation 2, we can express t2 in terms of x: t2 = 12 km / (x - 3) km/h

Substituting these expressions for t1 and t2 into Equation 3, we get: 10 km / (x + 3) km/h + 12 km / (x - 3) km/h = 2 hours

To simplify the equation, we can multiply both sides by (x + 3)(x - 3) to eliminate the denominators: 10(x - 3) + 12(x + 3) = 2(x + 3)(x - 3)

Expanding and simplifying the equation, we get: 10x - 30 + 12x + 36 = 2(x^2 - 9)

Simplifying further, we have: 22x + 6 = 2x^2 - 18

Rearranging the equation, we get a quadratic equation: 2x^2 - 22x - 24 = 0

We can solve this quadratic equation to find the possible values of x.

Solving the Quadratic Equation

We can solve the quadratic equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula to find the values of x.

The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / (2a)

For our quadratic equation 2x^2 - 22x - 24 = 0, the coefficients are: a = 2, b = -22, and c = -24.

Substituting these values into the quadratic formula, we get: x = (-(-22) ± √((-22)^2 - 4(2)(-24))) / (2(2))

Simplifying further, we have: x = (22 ± √(484 + 192)) / 4

x = (22 ± √676) / 4

x = (22 ± 26) / 4

This gives us two possible values for x: x1 = (22 + 26) / 4 = 48 / 4 = 12 km/h x2 = (22 - 26) / 4 = -4 / 4 = -1 km/h

Since the speed of the river cannot be negative, we can discard the negative value of x.

Answer

Therefore, the speed of the river is 12 km/h.