Question

Из пункта А в пункт В, расстояние между которыми 60 км, одновременно выехали автомобилист и велосипедист. Известно, что в час автомобилист проезжает на 30 км больше, чем велосипедист. Определите скорость велосипедиста, если известно, что он прибыл в пункт В на 2 часа 40 минут позже автомобилиста. Ответ дайте в км/ч.

Answer 1

Пусть скорость велосипеда x км/ч.
Тогда скорость автомобиля x+30 км/ч
велосипедист был в пути 60/x часов, а автомобилист 60/(x+30) часов
2 часа 40 мин =2 2/3 часа =8/3 часа
Составим уравнение

60/x =60/(x+30)+8/3
60*3*(x+30) = 60*3x+8*x*(x+3)
15*3*(x+30)=15*3x+2*x*(x+30)
45x+1350 = 45x+2x^2+60x
2x^2+60x-1350=0
x^2+30x-675=0
D=30^2-4*1*(-675)=900+2700=3600
√D=60
x1=(-30+60)/2=30/2 =15 
x2=(-30-60)/2=-45<0 - не имеет физического смысла
Ответ: скорость велосипедиста 15 км/ч

Answer 2

Problem Analysis

We are given that there are two travelers, an automobilist and a cyclist, who simultaneously start their journey from point A to point B. The distance between points A and B is 60 km. We also know that the automobilist travels 30 km more per hour than the cyclist. Additionally, we are given that the cyclist arrives at point B 2 hours and 40 minutes later than the automobilist. We need to determine the speed of the cyclist in km/h.

Solution

Let's assume the speed of the cyclist is x km/h. Since the automobilist travels 30 km more per hour than the cyclist, the speed of the automobilist will be x + 30 km/h.

We can use the formula distance = speed × time to calculate the time taken by each traveler to reach point B.

The time taken by the cyclist to travel from point A to point B is given by:

time taken by cyclist = distance / speed of cyclist

Similarly, the time taken by the automobilist to travel from point A to point B is given by:

time taken by automobilist = distance / speed of automobilist

We are given that the cyclist arrives at point B 2 hours and 40 minutes later than the automobilist. We can convert 2 hours and 40 minutes to hours by dividing it by 60:

2 hours and 40 minutes = 2 + 40/60 = 2.67 hours

Using the above information, we can set up the following equation:

time taken by cyclist = time taken by automobilist + 2.67

Substituting the values from the previous equations, we get:

distance / speed of cyclist = distance / speed of automobilist + 2.67

Since the distance is the same for both travelers, we can cancel it out:

speed of automobilist = speed of cyclist + 30

Now we have a system of two equations:

1. speed of automobilist = speed of cyclist + 30 2. speed of cyclist = distance / (speed of automobilist) + 2.67

We can solve this system of equations to find the speed of the cyclist.

Let's substitute the value of the speed of automobilist from equation 1 into equation 2:

speed of cyclist = distance / ((speed of cyclist + 30)) + 2.67

Simplifying the equation:

speed of cyclist × (speed of cyclist + 30) = distance + 2.67 × (speed of cyclist + 30)

Expanding and rearranging the equation:

speed of cyclist^2 + 30 × speed of cyclist = distance + 2.67 × speed of cyclist + 2.67 × 30

speed of cyclist^2 + 30 × speed of cyclist - distance - 2.67 × speed of cyclist - 2.67 × 30 = 0

The equation is now in the form ax^2 + bx + c = 0, where a = 1, b = 30, and c = -distance - 2.67 × 30.

We can solve this quadratic equation to find the speed of the cyclist.

Calculation

Given that the distance between points A and B is 60 km, we can substitute this value into the equation:

c = -60 - 2.67 × 30 = -60 - 80.1 = -140.1

Now we have the quadratic equation:

speed of cyclist^2 + 30 × speed of cyclist - 140.1 = 0

We can solve this equation using the quadratic formula:

speed of cyclist = (-b ± √(b^2 - 4ac)) / (2a)

Substituting the values into the formula:

speed of cyclist = (-(30) ± √((30)^2 - 4(1)(-140.1))) / (2(1))

Simplifying the equation:

speed of cyclist = (-30 ± √(900 + 560.4)) / 2

speed of cyclist = (-30 ± √(1460.4)) / 2

Calculating the square root:

speed of cyclist = (-30 ± 38.18) / 2

Now we have two possible values for the speed of the cyclist:

1. speed of cyclist = (-30 + 38.18) / 2 = 8.18 / 2 = 4.09 km/h 2. speed of cyclist = (-30 - 38.18) / 2 = -68.18 / 2 = -34.09 km/h

Since speed cannot be negative in this context, we can discard the negative value.

Answer

The speed of the cyclist is 4.09 km/h.

Conclusion

In this problem, we used the given information about the distance, speed difference, and time difference between an automobilist and a cyclist to determine the speed of the cyclist. By setting up and solving a system of equations, we found that the speed of the cyclist is 4.09 km/h.