Question

Из пункта А в пункт В, расстояние между которыми равно 18 км, вышел пешеход. Через два часа за ним выехал велосипедист, который проезжал за каждый час на 4,5 км больше, чем проходил пешеход. Определить скорость движения велосипедиста, если известно, что прибыл в пункт В одновременно с пешеходом.

Answer 1

Х -скорость пешехода, время его пути 18/х 
х+4,5 - скорость велосипедиста, время его пути - 18/(х+4,5)
Уравнение
18/х-18/(х+4,5)=2
9(х+4,5)-9х=х(х+4,5)
40,5=х²+4,5х
х²+4,5х-40,5=0
D=4.5²+4*40.5=182.25=13.5²
x=(-4.5+13.5)/2=4.5 км/ч - скорость пешехода
4,5+4,5=9км/ч - скорость велосипедиста

Answer 2

Problem Analysis

We are given that a pedestrian starts walking from point A to point B, which are 18 km apart. After two hours, a cyclist starts riding from point A to point B, and for every hour, the cyclist covers 4.5 km more than the pedestrian. We need to determine the speed of the cyclist if they arrive at point B at the same time as the pedestrian.

Solution

Let's assume the speed of the pedestrian is x km/h. Since the pedestrian walks for 2 hours, the distance covered by the pedestrian is 2x km.

The speed of the cyclist is 4.5 km/h more than the pedestrian's speed, so the speed of the cyclist is x + 4.5 km/h. The cyclist rides for an unknown number of hours, which we'll call t.

The distance covered by the cyclist is the product of the speed and time: (x + 4.5) * t km.

According to the problem, the total distance covered by both the pedestrian and the cyclist is 18 km. Therefore, we can write the equation:

2x + (x + 4.5) * t = 18

We need to solve this equation to find the value of t and then calculate the speed of the cyclist.

Solving the Equation

Let's solve the equation step by step:

2x + (x + 4.5) * t = 18

Expanding the equation:

2x + xt + 4.5t = 18

Rearranging the terms:

xt + 2x + 4.5t = 18

Factoring out x:

x(t + 2) + 4.5t = 18

Now, we can solve for t:

x(t + 2) + 4.5t = 18

xt + 2x + 4.5t = 18

t(x + 4.5) + 2x = 18

t(x + 4.5) = 18 - 2x

t = (18 - 2x) / (x + 4.5)

Now that we have the value of t, we can substitute it back into the equation to find the speed of the cyclist:

Speed of the cyclist = x + 4.5

Speed of the cyclist = x + 4.5 = x + 4.5 = (18 - 2x) / (x + 4.5) + 4.5

Simplifying the equation:

x + 4.5 = (18 - 2x) / (x + 4.5) + 4.5

x + 4.5 = (18 - 2x + 4.5(x + 4.5)) / (x + 4.5)

x + 4.5 = (18 - 2x + 4.5x + 20.25) / (x + 4.5)

x + 4.5 = (18 + 2.5x + 20.25) / (x + 4.5)

x + 4.5 = (38.25 + 2.5x) / (x + 4.5)

Cross-multiplying:

(x + 4.5)(x + 4.5) = 38.25 + 2.5x

Expanding and simplifying:

x^2 + 9x + 20.25 = 38.25 + 2.5x

x^2 + 9x - 2.5x + 20.25 - 38.25 = 0

x^2 + 6.5x - 18 = 0

Using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Where a = 1, b = 6.5, and c = -18.

Solving for x:

x = (-6.5 ± √(6.5^2 - 4 * 1 * -18)) / (2 * 1)

x = (-6.5 ± √(42.25 + 72)) / 2

x = (-6.5 ± √(114.25)) / 2

x = (-6.5 ± 10.69) / 2

Taking the positive value:

x = (-6.5 + 10.69) / 2

x = 4.19 / 2

x = 2.095

Therefore, the speed of the pedestrian is approximately 2.095 km/h.

Now, we can calculate the speed of the cyclist:

Speed of the cyclist = x + 4.5

Speed of the cyclist = 2.095 + 4.5

Speed of the cyclist ≈ 6.595 km/h

Therefore, the speed of the cyclist is approximately 6.595 km/h.

Answer

The speed of the cyclist is approximately 6.595 km/h.