Question

Помогите решить задачу. Велосипедист собирался проехать дистанцию 120км с кое-какой скоростью.Но он ехал со скоростью на 6 км в час меньше запланированной и приехал на 1час позже, чем планировал. С какой скоростью он ехал?

Answer 1

Х - реальная скорость велосипедиста.
(х + 6) - предполагаемая скорость велосипедиста
120:х -реальное время в пути
120:(х + 6) - предполагаемое время, которое на 1 час меньше.
Уравнение:
120:х - 120:(х + 6) = 1
120·(х + 6) - 120х = х·(х + 6)
120х + 720 - 120х = х² + 6х
х² + 6х - 720 = 0
D = 36 + 2880 = 2916
√D = 54
х₁ = (-6 - 54):2 = -30 не подходит, т.к. скорость не может быть отрицательной
х₁ = (-6 + 54):2 = 24
Ответ: реальная скорость велосипедиста равна 24км/ч

Answer 2

Problem Analysis

We are given that a cyclist planned to travel a distance of 120 km at a certain speed. However, the cyclist ended up traveling at a speed 6 km/h slower than planned and arrived 1 hour later than expected. We need to determine the speed at which the cyclist was traveling.

Solution

Let's assume the planned speed of the cyclist was x km/h. Since the cyclist traveled at a speed 6 km/h slower than planned, the actual speed can be represented as (x - 6) km/h.

We can use the formula distance = speed × time to solve this problem. The time taken to travel the distance of 120 km at the planned speed x km/h is given by 120 / x hours. Similarly, the time taken to travel the same distance at the actual speed (x - 6) km/h is given by 120 / (x - 6) hours.

According to the problem, the actual time taken was 1 hour more than the planned time. So, we can set up the equation:

120 / (x - 6) = 120 / x + 1

To solve this equation, we can cross-multiply and simplify:

120x = 120(x - 6) + x(x - 6)

Simplifying further:

120x = 120x - 720 + x^2 - 6x

Combining like terms:

0 = x^2 - 6x - 720

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

Solving the Quadratic Equation

To solve the quadratic equation x^2 - 6x - 720 = 0, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = -6, and c = -720. Substituting these values into the quadratic formula:

x = (-(-6) ± √((-6)^2 - 4(1)(-720))) / (2(1))

Simplifying further:

x = (6 ± √(36 + 2880)) / 2

x = (6 ± √(2916)) / 2

x = (6 ± 54) / 2

We have two possible solutions:

1. x = (6 + 54) / 2 = 60 / 2 = 30 2. x = (6 - 54) / 2 = -48 / 2 = -24

Since the speed cannot be negative, we can conclude that the cyclist was traveling at a speed of 30 km/h.

Answer

The cyclist was traveling at a speed of 30 km/h.

Verification

To verify this answer, we can substitute the value of x into the equation:

120 / (x - 6) = 120 / x + 1

Substituting x = 30:

120 / (30 - 6) = 120 / 30 + 1

Simplifying:

120 / 24 = 4 + 1

5 = 5

The equation holds true, which confirms that the cyclist was traveling at a speed of 30 km/h.