Question

Моторная лодка прошла 63 км по течению реки и 45 км про¬тив течения, затратив на весь путь 6 ч. Найдите скорость лодки в стоячей воде и скорость течения, если известно, что, двигаясь 5 ч по течению реки, она проходит тот же путь, что за 7 ч против течения.

Answer 1

Пусть х км/ч - собственная скорость лодки в стоячей воде,
у км/ч - скорость течения реки,
Значит, (х+у) км/ч - скорость лодки по течению реки,
             (х-у) км/ч - скорость лодки против течения реки.
По условию задачи, известно, что лодка, за 5 ч по течению, прошла тот же путь, что за 7 часов против течения реки.
Составляем уравнение:
5(x+y)=7(x-y)
5x+5y=7x-7y
5y+7y=7x-5x
12y=2x
6y=x
Итак, х+у=6у+у=7у - скорость лодки по течению реки,
          х-у =6у-у=5у - скорость лодки против течения реки.
Тогда 63/7у = 9/у час - время лодки на движение по течению реки,
          45/5у =9/у  час - время лодки на движение против течения реки.
По условию задачи, на весь путь лодка затратила 6 часов.
Составим уравнение:
9/у + 9/у = 6
(2*9)/у=6
18/у=6
у=18/6
у=3 (км/ч) - скорость  течения реки
х=6*3=18 (км/ч) - собственная скорость лодки

Answer 2

Problem Analysis

We are given that a motorboat traveled 63 km downstream and 45 km upstream, taking a total of 6 hours for the entire journey. We are asked to find the speed of the boat in still water and the speed of the current.

Downstream Speed Calculation

Let's assume the speed of the boat in still water is B km/h and the speed of the current is C km/h. When the boat is traveling downstream, its effective speed is the sum of its speed in still water and the speed of the current. Therefore, the downstream speed of the boat is B + C km/h.

Upstream Speed Calculation

When the boat is traveling upstream, its effective speed is the difference between its speed in still water and the speed of the current. Therefore, the upstream speed of the boat is B - C km/h.

Time Calculation

We are given that the boat takes 6 hours for the entire journey. We can set up the following equation based on the given information:

Downstream time + Upstream time = Total time

The downstream time can be calculated by dividing the distance traveled downstream by the downstream speed:

63 km / (B + C) km/h

The upstream time can be calculated by dividing the distance traveled upstream by the upstream speed:

45 km / (B - C) km/h

Substituting these values into the equation, we get:

63 / (B + C) + 45 / (B - C) = 6

Solving the Equation

To solve the equation, we can simplify it by multiplying both sides by the least common multiple (LCM) of the denominators, which is (B + C)(B - C). This will eliminate the denominators and allow us to solve for B and C.

After simplification, the equation becomes:

63(B - C) + 45(B + C) = 6(B + C)(B - C)

Simplifying further, we get:

63B - 63C + 45B + 45C = 6(B^2 - C^2)

Combining like terms, we have:

108B - 18C = 6B^2 - 6C^2

Rearranging the equation, we get:

6B^2 - 108B + 6C^2 - 18C = 0

Dividing both sides by 6, we get:

B^2 - 18B + C^2 - 3C = 0

Solving the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

B = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = -18, and c = -3.

Substituting these values into the quadratic formula, we get:

B = (-(-18) ± √((-18)^2 - 4(1)(-3))) / (2(1))

Simplifying further, we have:

B = (18 ± √(324 + 12)) / 2

B = (18 ± √336) / 2

B = (18 ± 4√21) / 2

Simplifying the expression, we get:

B = 9 ± 2√21

Since the speed of the boat cannot be negative, we can discard the negative solution:

B = 9 + 2√21

Finding the Speed of the Current

To find the speed of the current, we can substitute the value of B into one of the original equations. Let's use the downstream equation:

B + C = 63 / t

Substituting the value of B and the given downstream time of 5 hours, we get:

9 + 2√21 + C = 63 / 5

Simplifying further, we have:

9 + 2√21 + C = 12.6

Subtracting 9 and 2√21 from both sides, we get:

C = 12.6 - 9 - 2√21

C = 3.6 - 2√21

Therefore, the speed of the current is approximately 3.6 - 2√21 km/h.

Summary

Based on the given information, the speed of the boat in still water is approximately 9 + 2√21 km/h, and the speed of the current is approximately 3.6 - 2√21 km/h.

Please note that the values provided are approximate due to the use of the quadratic formula and rounding.