Question

Моторная лодка прошла 48 км по течению реки и вернулась обратно,потратив на весь путь 7 часов. Скорость течения реки равна 2 км/ч.Найдите скорость лодки в неподвижной воде.

Answer 1

х км/ч скорость лодки в неподвижной воде,

(х+2) км/ч скорость движения лодки по течению реки

(х-2) км/ч скорость движения лодки против течения реки; на весь путь по течению затрачено 48/(х+2) ч, а против течения 48/(х-2) ч. Согласно условию задачи имеем уравнение 48/(х+2) + 48/(х-2) = 7. Решая его получаем 48(х-2)+48(х+2)=7(х в квадрате - 4), после преобразований, получим 96х-7х в квадрате +28=0, или

7х в квадрате -96х -28=0, решаем квадратное уравнение, D= 96*96 -4*7*(-28)=10000

х1=(96-квадратный корень из дискриминанта (это есть 100))/2*7=-4/14=-2/7;

х2=(96+100)/14=14. Условию задачи удовлетворяет положительный ответ. Итак, скорость лодки в неподвижной воде 14 км/ч.

Answer 2

Пусть х км/ч скорость лодки в стоячей воде ( т.е. без течения ) тогда скорость по течению х+2, а против х-2. Весь путь составил 7 ч. Составляю уравнение

48/(х+2) + 48/(х-2) = 7

48х-96+48х+96-7(х^2-4)=0

96х-7х^2+28=0

-7х^2+96х+28=0

Д=9216+784=100^2

х1=(-96-100)/-14=-196/-14=14

х2=(-96+100)/-14 нет решений

Ответ: скорость в неподвижной воде 14 км/ч

Answer 3

Problem Analysis

We are given that a motorboat traveled 48 km upstream on a river and then returned back, taking a total of 7 hours for the entire trip. The speed of the river's current is given as 2 km/h. We need to find the speed of the motorboat in still water.

Solution

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

When the motorboat is traveling upstream against the current, its effective speed is reduced by the speed of the current. So, the speed of the motorboat relative to the ground is (x - 2) km/h.

When the motorboat is traveling downstream with the current, its effective speed is increased by the speed of the current. So, the speed of the motorboat relative to the ground is (x + 2) km/h.

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

Calculation

Let's calculate the time taken for each leg of the journey.

- Upstream journey: - Distance = 48 km - Speed = (x - 2) km/h - Time = distance / speed = 48 / (x - 2) hours

- Downstream journey: - Distance = 48 km - Speed = (x + 2) km/h - Time = distance / speed = 48 / (x + 2) hours

According to the problem, the total time taken for the entire trip is 7 hours. So, we can write the equation:

48 / (x - 2) + 48 / (x + 2) = 7

To solve this equation, we can multiply both sides by (x - 2)(x + 2) to eliminate the denominators:

48(x + 2) + 48(x - 2) = 7(x - 2)(x + 2)

Simplifying the equation:

48x + 96 + 48x - 96 = 7(x^2 - 4)

96x = 7x^2 - 28

Rearranging the equation:

7x^2 - 96x - 28 = 0

Now, we can solve this quadratic equation to find the value of x.

Quadratic Equation Solution

Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a), where a = 7, b = -96, and c = -28.

Substituting the values:

x = (-(-96) ± √((-96)^2 - 4 * 7 * (-28))) / (2 * 7)

Simplifying:

x = (96 ± √(9216 + 784)) / 14

x = (96 ± √(9999)) / 14

Since we are looking for the speed of the motorboat, the answer should be positive. Therefore, we can discard the negative solution.

Calculating the positive solution:

x = (96 + √(9999)) / 14

x ≈ 14.68 km/h

Answer

The speed of the motorboat in still water is approximately 14.68 km/h.