Question

на путь, который равен 2 км, велосипедист потратил на 12 мин меньше, чем пешеход, так как его скорость на 9 км/ч больше чем у пешехода. Найдите скорости велосипедиста и пешехода.

Answer 1

Время движения велосипедиста: t1 = s/v = 2/v. Переведем 12 минут в часы. Если 60 мин - это 1 час, то 12 минут - это 12/60 = 0.2 (часа).

Время движения пешехода: t2 = s/v = 2/(v-9). Велосипедист потратил на преодоление пути на 0.2 часа меньше, то есть:

t2 - t1 = 2/(v-9) - 2/v = 0.2.

Умножим обе части полученного равенства на 5:

10/(v-9) - 10/v = 1;

(10v-10v+90)/(v(v-9)) = 1;

v^2 - 9v - 90 = 0; (*)

Используем теорему, обратной теореме Виета: сумма корней уравнения (*) равна 9, произведение - -90. Очевидно, что корни - это числа 15 и -6.

Скорость не может быть выражена отрицательным числом, следовательно, скорость движения велосипедиста равна 15 км/ч, пешехода - (15-9) км/ч = 6 км/ч.

Ответ: 15 км/ч; 6 км/ч.

Answer 2

Problem Analysis

We are given that a cyclist traveled a distance of 2 km and took 12 minutes less than a pedestrian to complete the journey. The cyclist's speed is 9 km/h faster than the pedestrian's speed. We need to find the speeds of the cyclist and the pedestrian.

Solution

Let's assume the speed of the pedestrian is x km/h. Since the cyclist's speed is 9 km/h faster, the cyclist's speed can be represented as x + 9 km/h.

We know that speed is equal to distance divided by time. Let's calculate the time taken by the cyclist and the pedestrian to travel 2 km.

The time taken by the pedestrian can be calculated using the formula: time = distance / speed.

For the pedestrian: time_pedestrian = 2 / x hours.

The time taken by the cyclist is 12 minutes (or 0.2 hours) less than the time taken by the pedestrian: time_cyclist = time_pedestrian - 0.2 hours.

Now, we can calculate the time taken by the cyclist: time_cyclist = (2 / x) - 0.2 hours.

Since we know that time is equal to distance divided by speed, we can set up the following equation for the cyclist: (2 / (x + 9)) = (2 / x) - 0.2.

To solve this equation, we can cross-multiply and simplify: 2x = 2(x + 9) - 0.2x(x + 9).

Simplifying further: 2x = 2x + 18 - 0.2x^2 - 1.8x.

Combining like terms: 0 = -0.2x^2 - 1.8x + 18.

Now, we have a quadratic equation. We can solve it using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).

For our equation: a = -0.2, b = -1.8, and c = 18.

Plugging in the values and solving for x, we can find the speed of the pedestrian. Once we have the speed of the pedestrian, we can calculate the speed of the cyclist by adding 9 km/h.

Let's calculate the speeds of the cyclist and the pedestrian.

Calculation

Using the quadratic formula, we can solve for x:

x = (-(-1.8) ± √((-1.8)^2 - 4(-0.2)(18))) / (2(-0.2)).

Simplifying further: x = (1.8 ± √(3.24 + 14.4)) / (-0.4).

x = (1.8 ± √(17.64)) / (-0.4).

x = (1.8 ± 4.2) / (-0.4).

There are two possible solutions for x:

1. x = (1.8 + 4.2) / (-0.4) = 15 km/h. 2. x = (1.8 - 4.2) / (-0.4) = -5 km/h.

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

Therefore, the speed of the pedestrian is 15 km/h.

The speed of the cyclist is 15 + 9 = 24 km/h.

Answer

The speed of the pedestrian is 15 km/h, and the speed of the cyclist is 24 km/h.

Note: The negative solution for the speed of the pedestrian is discarded because speed cannot be negative in this context.