Question

задача. моторная лодка прошла 100 км по озеру и 4 км против течения реки , затратив на весь путь 1 час . Найти собственную скорость лодки , если скорость течения реки равна 3 км\ч

Answer 1

Пусть х км/ч -собственная скорость лодки, т.е. по озеру, тогда  (х-3) км/ч - скорость лодки против течения. 100:х - время, затраченное на движение по озеру, 4:(х-3) - время движения против течения. Всего было потрачено 1 ч. Составляем уравнение:
100/х + 4/(х-3) = 1

 

Answer 2

Problem Analysis

We are given that a motorboat traveled 100 km on a lake and then 4 km against the current of a river, taking a total of 1 hour for the entire journey. We need to find the speed of the boat if the speed of the river current is 3 km/h.

Solution

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

When the motorboat is traveling on the lake, it is not affected by the current of the river. Therefore, the time taken to travel 100 km on the lake is given by:

Time on lake = Distance / Speed = 100 km / x km/h = 100/x hours

When the motorboat is traveling against the current of the river, its effective speed is reduced by the speed of the river current. Therefore, the time taken to travel 4 km against the current is given by:

Time against current = Distance / (Speed - Current) = 4 km / (x km/h - 3 km/h) = 4/(x-3) hours

The total time taken for the entire journey is given as 1 hour. Therefore, we can write the equation:

Time on lake + Time against current = Total time

Substituting the values, we get:

100/x + 4/(x-3) = 1

To solve this equation, we can multiply through by x(x-3) to eliminate the denominators:

100(x-3) + 4x = x(x-3)

Simplifying the equation:

100x - 300 + 4x = x^2 - 3x

Rearranging the equation:

x^2 - 107x + 300 = 0

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

Quadratic Equation Solution

Using the quadratic formula, we have:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation x^2 - 107x + 300 = 0, the values of a, b, and c are:

a = 1 b = -107 c = 300

Substituting these values into the quadratic formula, we get:

x = (-(-107) ± √((-107)^2 - 4(1)(300))) / (2(1))

Simplifying further:

x = (107 ± √(11449 - 1200)) / 2

x = (107 ± √10249) / 2

x = (107 ± 101.24) / 2

We have two possible solutions:

x1 = (107 + 101.24) / 2 = 208.24 / 2 = 104.12 km/h x2 = (107 - 101.24) / 2 = 5.76 / 2 = 2.88 km/h

Since the speed of the motorboat cannot be negative, we can discard the second solution.

Answer

The speed of the motorboat is 104.12 km/h.

Verification

Let's verify our answer by substituting the value of x into the equation:

100/x + 4/(x-3) = 1

Substituting x = 104.12 km/h:

100/104.12 + 4/(104.12-3) = 1

0.9607 + 0.0393 = 1

1 = 1

The equation holds true, confirming that our answer is correct.

Conclusion

The speed of the motorboat is 104.12 km/h.