Question

Из А в В одновременно выехали два автомобилиста. Первый проехал с постоянной скоростью весь путь. Второй проехал первую половину пути со скоростью, меньшей скорости первого на 15 км/ч, а вторую половину пути — со скоростью 90 км/ч, в результате чего прибыл в В одновременно с первым автомобилистом.Найдите скорость первого автомобилиста, если известно, что она больше 50 км/ч. Ответ дайте в км/ч.

Answer 1

Х км/ч - скорость І
(х-15) км/ч - скорость ІІ на первой половине пути
Время движения одинаковое
t=S:V
1 - весь путь
1/х = 0,5/(х-15) + 0,5/90
Общий знаменатель - 90х(х-15)
90х-1350=45х+0,5х²-7,5х
0,5х²-52,5х+1350=0  (сокращаем на 0,5)
х²-105х+2700=0
D=b²-4ac
D=11025-10800=225
х1=(105+15)/2
х1=60(км/ч) - скорость І автомобилиста
х2=(105-15)/2=45(км/ч) - скорость ІІ

Answer 2

Problem Analysis

We are given that two drivers, A and B, simultaneously set off from point A to point B. Driver A travels the entire distance at a constant speed, while driver B travels the first half of the distance at a speed 15 km/h slower than driver A and the second half at a speed of 90 km/h. We need to find the speed of driver A, given that it is greater than 50 km/h.

Solution

Let's assume that the total distance from point A to point B is D km.

We can divide the journey into two parts: the first half and the second half.

For the first half of the journey, driver B travels at a speed 15 km/h slower than driver A. Let's call the speed of driver A as V km/h. Therefore, the speed of driver B for the first half of the journey is (V - 15) km/h.

For the second half of the journey, driver B travels at a speed of 90 km/h.

Since both drivers arrive at point B simultaneously, we can set up the following equation:

Time taken by driver A for the entire journey = Time taken by driver B for the first half of the journey + Time taken by driver B for the second half of the journey

To calculate the time taken by each driver, we can use the formula:

Time = Distance / Speed

For driver A: Time taken by driver A for the entire journey = D / V

For driver B: Time taken by driver B for the first half of the journey = (D/2) / (V - 15) Time taken by driver B for the second half of the journey = (D/2) / 90

Setting up the equation: D / V = (D/2) / (V - 15) + (D/2) / 90

Simplifying the equation: D / V = (D/2) * [(1 / (V - 15)) + (1 / 90)]

Now, we can solve this equation to find the value of V.

Calculation

Let's solve the equation to find the value of V.

D / V = (D/2) * [(1 / (V - 15)) + (1 / 90)]

Multiplying both sides by V: D = (D/2) * [(V / (V - 15)) + (V / 90)]

Multiplying both sides by 2: 2D = D * [(V / (V - 15)) + (V / 90)]

Simplifying the equation: 2 = (V / (V - 15)) + (V / 90)

Multiplying both sides by (V - 15) * 90: 2 * (V - 15) * 90 = V * 90 + V * (V - 15)

Expanding and simplifying: 180V - 2700 = 90V + V^2 - 15V

Combining like terms: V^2 + 75V - 2700 = 0

Now, we can solve this quadratic equation to find the value of V.

Using the quadratic formula: V = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = 75, and c = -2700.

Calculating the discriminant: Discriminant = b^2 - 4ac = 75^2 - 4(1)(-2700) = 5625 + 10800 = 16425

Since the discriminant is positive, we have two real solutions for V.

Calculating the solutions: V1 = (-75 + √16425) / 2 ≈ 45.47 km/h V2 = (-75 - √16425) / 2 ≈ -120.47 km/h

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

Answer

The speed of the first driver, driver A, is approximately 45.47 km/h.