Question

моторная лодка на 21 км по течению реки и на обратный путь затратил 2 часа и 40 мин. В другой раз та же моторная лодка прошла по течению реки 18 км и против течения 14 затратив на весь путь 2 часа.Какова собственная скорость моторной лодки и скорость течения?

Answer 1

Problem Analysis

We are given two scenarios involving a motorboat traveling along a river. In the first scenario, the boat travels 21 km downstream and takes 2 hours and 40 minutes for the round trip. In the second scenario, the boat travels 18 km downstream and 14 km upstream, taking 2 hours for the round trip. We need to determine the speed of the motorboat and the speed of the river current.

Solution

Let's assume the speed of the motorboat is x km/h and the speed of the river current is y km/h.

In the first scenario, the boat travels 21 km downstream. The time taken for the downstream journey is given as 2 hours and 40 minutes, which is equivalent to 2.67 hours. The time taken for the upstream journey will be the same.

Using the formula distance = speed × time, we can write the following equations:

For the downstream journey: 21 = (x + y) × 2.67 For the upstream journey: 21 = (x - y) × 2.67 In the second scenario, the boat travels 18 km downstream and 14 km upstream. The total time taken for the round trip is 2 hours.

Using the same formula, we can write the following equations:

For the downstream journey: 18 = (x + y) × 2 For the upstream journey: 14 = (x - y) × 2 We now have a system of equations with four unknowns: x, y, t1, and t2. We need to solve this system of equations to find the values of x and y.

Solving the System of Equations

To solve the system of equations, we can use substitution or elimination. Let's use the elimination method.

From equations and we can subtract equation from equation to eliminate the y term:

21 - 21 = (x + y) × 2.67 - (x - y) × 2.67

Simplifying the equation, we get:

0 = 2xy × 2.67

Since the product of xy is zero, either x or y must be zero. However, the speed of the motorboat and the speed of the river current cannot be zero, so this is not a valid solution.

Let's move on to the second set of equations, and We can subtract equation from equation to eliminate the y term:

18 - 14 = (x + y) × 2 - (x - y) × 2

Simplifying the equation, we get:

4 = 4xy

Dividing both sides by 4, we get:

1 = xy

Now we have a valid equation. Since xy = 1, we can substitute this value into either equation or to solve for x or y.

Let's substitute xy = 1 into equation:

18 = (x + y) × 2

Dividing both sides by 2, we get:

9 = x + y

Now we have a new equation, 9 = x + y.

We can solve this equation together with xy = 1 using substitution or elimination to find the values of x and y.

Solving the New Equation

Let's solve the equation 9 = x + y together with xy = 1 using the substitution method.

From the equation xy = 1, we can express y in terms of x:

y = 1/x

Substituting this value into the equation 9 = x + y, we get:

9 = x + 1/x

Multiplying both sides by x, we get:

9x = x^2 + 1

Rearranging the equation, we get:

x^2 - 9x + 1 = 0

Now we have a quadratic equation. We can solve this equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = -9, and c = 1. Substituting these values into the quadratic formula, we get:

x = (-(-9) ± √((-9)^2 - 4(1)(1))) / (2(1))

Simplifying the equation, we get:

x = (9 ± √(81 - 4)) / 2

x = (9 ± √77) / 2

Now we have two possible values for x. Let's calculate them:

x1 = (9 + √77) / 2

x2 = (9 - √77) / 2

Since the speed of the motorboat cannot be negative, we can discard the negative value. Therefore, the speed of the motorboat is:

x = (9 + √77) / 2

Now that we have the value of x, we can substitute it back into the equation 9 = x + y to solve for y:

9 = (9 + √77) / 2 + y

Multiplying both sides by 2, we get:

18 = 9 + √77 + 2y

Subtracting 9 and √77 from both sides, we get:

9 - √77 = 2y

Dividing both sides by 2, we get:

y = (9 - √77) / 2

Therefore, the speed of the motorboat is (9 + √77) / 2 km/h and the speed of the river current is (9 - √77) / 2 km/h.

Answer

The speed of the motorboat is (9 + √77) / 2 km/h and the speed of the river current is (9 - √77) / 2 km/h.

Note: The above solution assumes that the boat maintains a constant speed throughout the journey and that the river current is uniform.