Question

Відстань від пункту А до пункту В по шосе дорівнює 180 км, а по залізниці – 210 км. Автомобіль з пункту А виїхав на 30 хв. пізніше, ніж поїзд, і прибув до пункту В на 45 хв. раніше. Знайти швидкість автомобіля, якщо вона на 20 км/год більша швидкості поїзда

Answer 1

Х км/ч - скорость поезда
х + 20 км/ч - скорость автомобиля
30 + 45 = 75 минут = на 1,25 часа дольше был в пути поезд,чем автомобиль
210/х часов - был в пути поезд
180/(х + 20) часов - был в пути автомобиль

$\frac{180}{x+20}+1,25= \frac{210}{x} \\ \\ 180x+1,25x^2+25x=210x+4200 \\ \\ 1,25x^2-5x-4200=0 \\ \\ x^2-4x-3360=0$

По теореме Виета: х₁ = -56
                                х₂ = 60

х₁ = -56 - не удовлетворяет условию задачи

60 км/ч - скорость поезда
60 + 20 = 80 км/ч - скорость автомобиля

Answer 2

Problem Analysis

We are given that the distance from point A to point B is 180 km by road and 210 km by rail. We are also given that a car left point A 30 minutes later than a train and arrived at point B 45 minutes earlier. We need to find the speed of the car, given that it is 20 km/h faster than the speed of the train.

Solution

Let's assume the speed of the train is x km/h. Therefore, the speed of the car is x + 20 km/h.

We can start by calculating the time it takes for the train to travel from point A to point B by rail. We can use the formula time = distance / speed. In this case, the distance is 210 km and the speed is x km/h. Therefore, the time taken by the train is 210 / x hours.

Similarly, we can calculate the time it takes for the car to travel from point A to point B by road. The distance is 180 km and the speed is x + 20 km/h. Therefore, the time taken by the car is 180 / (x + 20) hours.

According to the given information, the car left 30 minutes (or 0.5 hours) later than the train and arrived 45 minutes (or 0.75 hours) earlier. Therefore, the total time taken by the car is the time taken by the train minus the time difference.

Let's set up the equation:

210 / x - 0.5 = 180 / (x + 20) + 0.75

Now, we can solve this equation to find the value of x, which represents the speed of the train.

Calculation

Let's solve the equation step by step:

210 / x - 0.5 = 180 / (x + 20) + 0.75

Multiply both sides of the equation by x(x + 20) to eliminate the denominators:

**210(x + 20) - 0.5x(x + 20