Question

Велосипедист выехал с постоянной скоростью из города А в город В, расстояние между которыми равно 180км. Отдохнув, он отправился обратно в А, увеличив скорость на 5 км/ч. По пути он сделал остановку на 3 часа, в результате чего затратил на обратный путь столько же времени, сколько на путь из А в В. Найдите скорость велосипедиста на пути из А в В

Answer 1

Пусть х км/ч - скорость велосипедиста из А в Б,
тогда x+5 км/ч - будет его скорость из Б в А
180/x ч. - время, за которое велосипедист прошел путь из А в Б
180/x+5 + 3 ч. - время, за которое велосипедист прошел путь из Б в А

Составим уравнение:
180/x+5 + 3= 180/x
180x+ 3x² + 15x=180x+900
3x²+15x - 900=0
x²+5x-300=0
D= b²-4ac= 25+1200=1225 √1225 = 35
x1 = -b - √D / 2a = -5 - 35/2 = -40/2 = -20 (не подходит по смыслу задачи)
x2 = -b + √D / 2a = -5+35/2=30/2=15

Ответ:15 км/ч

Answer 2

Составим уравнение:
180/x+5 + 3= 180/x
180x+ 3x² + 15x=180x+900
3x²+15x - 900=0
x²+5x-300=0
D= b²-4ac= 25+1200=1225 √1225 = 35
x1 = -b - √D / 2a = -5 - 35/2 = -40/2 = -20 (не подходит по смыслу задачи)
x2 = -b + √D / 2a = -5+35/2=30/2=15

Ответ:15 км/ч

Answer 3

Problem Analysis

A cyclist travels from city A to city B at a constant speed. After resting, the cyclist returns from city B to city A, increasing their speed by 5 km/h. During the return journey, the cyclist makes a 3-hour stop, which results in the return journey taking the same amount of time as the journey from city A to city B. We need to find the speed of the cyclist on the journey from city A to city B.

Solution

Let's assume the speed of the cyclist on the journey from city A to city B is x km/h.

On the journey from city A to city B, the distance is 180 km, and the speed is x km/h. Therefore, the time taken for this journey is given by:

Time = Distance / Speed Time = 180 km / x km/h Time = 180 / x hours ---(1)

On the return journey from city B to city A, the distance is still 180 km, but the speed is (x + 5) km/h (increased by 5 km/h). The cyclist makes a 3-hour stop during this journey. Therefore, the time taken for this journey is given by:

Time = Distance / Speed + Stop Time Time = 180 km / (x + 5) km/h + 3 hours Time = 180 / (x + 5) + 3 hours ---(2)

According to the problem statement, the time taken for the return journey is the same as the time taken for the journey from city A to city B. Therefore, we can equate equations (1) and (2) and solve for x.

180 / x = 180 / (x + 5) + 3

Let's solve this equation to find the value of x.

Calculation

To solve the equation 180 / x = 180 / (x + 5) + 3 for x, we can follow these steps:

1. Multiply both sides of the equation by x(x + 5) to eliminate the denominators: 180(x + 5) = 180x + 3x(x + 5)

2. Expand and simplify the equation: 180x + 900 = 180x + 3x^2 + 15x

3. Rearrange the equation to form a quadratic equation: 3x^2 + 15x - 900 = 0

4. Divide the entire equation by 3 to simplify: x^2 + 5x - 300 = 0

5. Solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, we can factor the equation: (x + 20)(x - 15) = 0

6. Set each factor equal to zero and solve for x: - x + 20 = 0 --> x = -20 (extraneous solution, as speed cannot be negative) - x - 15 = 0 --> x = 15

Therefore, the speed of the cyclist on the journey from city A to city B is 15 km/h.

Answer

The speed of the cyclist on the journey from city A to city B is 15 km/h.