Question

Из А в В одновременно выехали два автомобилиста. Первый проехал с постоянной скоростью весь путь. Второй проехал первую половину пути со скоростью 30 км\ч, а вторую половину пути проехал со скоростью на 9 км\ч больше скорости первого, в результате чего прибыл в В одновременно с первым автомобилистом. Найдите скорость первого автомобилиста.

Answer 1

Пусть x - скорость первого автомобиля
t - за которое он проехал весь путь

Тогда имеем: x*t - весь путь

0.5*x*t/30 - время за которое второй автомобиль проехал первую половину пути

составляем уравнение второй части пути второго автомобиля
(t-0.5*x*t/30)*(x+9) = 0.5*x*t

(1-0.5x/30)*(x+9)=0.5*x
x-1/60*x^2+9-9/60x-0.5*x=0
x^2+9x-60x+30x-540=0
x^2-21x-540=0 
D=21^2+4*540=2601=51**2
x= (21+51)/2
x= 36 км - скорость первого автомобиля



  

Answer 2

Problem Analysis

We are given that two drivers, A and B, simultaneously left point A and traveled to point B. Driver A traveled the entire distance at a constant speed, while driver B traveled the first half of the distance at a speed of 30 km/h and the second half at a speed 9 km/h faster than driver A. We need to find the speed of driver A.

Solution

Let's assume the distance between point A and point B is d km.

Driver A traveled the entire distance at a constant speed, so the time taken by driver A is given by: time taken by A = distance / speed = d / speed_A ---(1)

Driver B traveled the first half of the distance at a speed of 30 km/h, so the time taken by driver B for the first half is given by: time taken by B for first half = distance / speed = (d/2) / 30 = d / 60 ---(2)

Driver B traveled the second half of the distance at a speed 9 km/h faster than driver A, so the time taken by driver B for the second half is given by: time taken by B for second half = distance / speed = (d/2) / (speed_A + 9) ---(3)

Since both drivers arrived at point B simultaneously, the total time taken by driver A should be equal to the total time taken by driver B. Therefore, we can write the equation: time taken by A = time taken by B for first half + time taken by B for second half ---(4)

Substituting equations (1), (2), and (3) into equation (4), we get: d / speed_A = d / 60 + (d/2) / (speed_A + 9)

Simplifying the equation, we can solve for speed_A: 1 / speed_A = 1 / 60 + 1 / (2 * (speed_A + 9))

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

Calculation

To solve the equation, we can multiply both sides by the least common multiple (LCM) of the denominators to eliminate the fractions.

The LCM of 60 and 2 * (speed_A + 9) is 60 * (speed_A + 9). Multiplying both sides of the equation by this LCM, we get: 60 * (speed_A + 9) / speed_A = 60 + 2

Simplifying the equation further: 60 * (speed_A + 9) = 60 * speed_A + 2 * speed_A

Expanding and simplifying: 60 * speed_A + 540 = 60 * speed_A + 2 * speed_A

Subtracting 60 * speed_A from both sides: 540 = 2 * speed_A

Dividing both sides by 2: speed_A = 270

Answer

The speed of the first driver, driver A, is 270 km/h.