Question

Велосипедист выехал с постоянной скоростью из города А в город В, расстояние между которыми равно 180 км. 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

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 trip taking the same amount of time as the trip from city A to city B. We need to find the cyclist's speed on the journey from city A to city B.

Solution

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

The distance between city A and city B is given as 180 km.

On the return journey, the cyclist increases their speed by 5 km/h, so the speed is now (x + 5) km/h.

We are given that the time taken for the return journey is the same as the time taken for the journey from city A to city B, after accounting for the 3-hour stop.

To find the cyclist's speed on the journey from city A to city B, we can set up the following equation:

Time taken for the journey from A to B = Time taken for the return journey

The time taken for the journey from A to B can be calculated using the formula:

Time = Distance / Speed

Let's calculate the time taken for the journey from A to B:

Time taken for the journey from A to B = 180 / x

The time taken for the return journey can be calculated using the formula:

Time = Distance / Speed

However, we need to account for the 3-hour stop. So the time taken for the return journey is:

Time taken for the return journey = (180 / (x + 5)) + 3

Setting up the equation:

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

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

Calculation

To solve the equation, we can start by multiplying both sides of the equation by x(x + 5) to eliminate the denominators:

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

Expanding and simplifying:

180x + 900 = 180x + 3x^2 + 15x

Rearranging the terms:

3x^2 + 15x - 900 = 0

Dividing the equation by 3 to simplify:

x^2 + 5x - 300 = 0

Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = 5, and c = -300. Substituting these values into the quadratic formula:

x = (-5 ± √(5^2 - 4(1)(-300))) / (2(1))

Simplifying:

x = (-5 ± √(25 + 1200)) / 2

x = (-5 ± √1225) / 2

x = (-5 ± 35) / 2

We have two possible solutions:

x1 = (-5 + 35) / 2 = 30 / 2 = 15

x2 = (-5 - 35) / 2 = -40 / 2 = -20

Since speed cannot be negative, we discard the negative solution.

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

Answer

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