Question

расстояние между городами А и В равно 420 км. Из города А в город В выехал автомобиль, а через 2 часа 30 минут следом за ним со скоростью 65 км/ч выехал мотоциклет , догнал автомобиль в городе С и повернул обратно . Когда он вернулся обратно в А, автомобиль прибыл в В.Найдите расстояние от А до С.Ответ дайте в километрах

Answer 1

х - скорость автомобиля

растояние от А до С длинее от В до С на 2,5*х км.

у растояние от В до С

$\left \{ {{y+(y+2.5x)=420} \atop {(y+2.5x)/65=y/x}} \right \left \{ {{2y+2.5x=420} \atop {(y+2.5x)*x=65y}} \right \left \{ {{2y=420-2.5x} \atop {yx+2.5x^2=65y}} \right \left \{ {{y=210-1.25x} \atop {210x-1.25x^2+2.5x^2=13650-81,25x}} \right$

 $\left \{ {{y=210-1.25x} \atop {291.25x+1.25x^2-13650=0}} \right D=b^2-4ac=291.25^2-(4*1.25*(-13650)=153076.5625;\ x_1=(-b+\sqrt{D})/(2a)=(-291.25+391,25)/2.5=40;\ x_2=(-b-\sqrt{D})/(2a)=(-291.25-391.25)/2.5=-273$

 Скорость не может быть отрецательной поэтому, скорость автомобиля 40 км/ч

у=210-1,25*40=160 от В до С

420-160=260 км

Ответ: от А до С 260 км

 

 

 

 

 

 

 

 

 

 

Answer 2

Problem Analysis

We are given that the distance between cities A and B is 420 km. A car starts from city A and after 2 hours and 30 minutes, a motorcycle starts from city A and catches up with the car in city C. The motorcycle then turns back and returns to city A, while the car continues to city B. We need to find the distance from city A to city C.

Solution

Let's break down the problem step by step:

1. The car travels for 2 hours and 30 minutes, which is equivalent to 2.5 hours. We don't know the speed of the car, so let's call it 'v' km/h. 2. The motorcycle catches up with the car in city C. We don't know the time it takes for the motorcycle to reach city C, so let's call it 't' hours. The speed of the motorcycle is given as 65 km/h. 3. The motorcycle turns back and returns to city A. The time it takes for the motorcycle to return to city A is also 't' hours. 4. The car continues from city C to city B. The time it takes for the car to reach city B is '2.5 + t' hours.

To find the distance from city A to city C, we need to find the value of 't' first. We can use the equation:

Distance = Speed × Time

For the car, the distance traveled is 'v × 2.5' km. For the motorcycle, the distance traveled is '65 × t' km.

Since the motorcycle catches up with the car in city C, the total distance traveled by both vehicles is equal to the distance between cities A and B, which is 420 km.

So we have the equation:

v × 2.5 + 65 × t = 420

Now we can solve this equation to find the value of 't'.

Calculation

Let's solve the equation:

v × 2.5 + 65 × t = 420

We don't have the value of 'v', so we can't solve for 't' directly. However, we can use the information given in the problem to find the value of 'v'.

We know that the car continues from city C to city B while the motorcycle returns to city A. The time it takes for the car to reach city B is '2.5 + t' hours. The distance traveled by the car is 'v × (2.5 + t)' km.

Since the total distance traveled by both vehicles is equal to the distance between cities A and B, we have the equation:

v × 2.5 + 65 × t + v × (2.5 + t) = 420

Simplifying the equation:

2.5v + 65t + 2.5v + vt = 420

5v + 65t + vt = 420

Now we have an equation with two variables, 'v' and 't'. We need another equation to solve for both variables.

From the information given in the problem, we know that the car travels for 2 hours and 30 minutes, which is equivalent to 2.5 hours. So we have another equation:

2.5v = 2.5 × v

Now we have a system of equations:

5v + 65t + vt = 420 2.5v = 2.5

We can solve this system of equations to find the values of 'v' and 't'.

Solution

Let's solve the system of equations:

5v + 65t + vt = 420 2.5v = 2.5

From the second equation, we can see that 'v' is equal to 1.

Substituting 'v' = 1 in the first equation:

5(1) + 65t + (1)t = 420

5 + 65t + t = 420

66t = 415

t ≈ 6.2879

Now we have the value of 't'. To find the distance from city A to city C, we can substitute 't' = 6.2879 in the equation:

v × 2.5 + 65 × t = 420

1 × 2.5 + 65 × 6.2879 = 420

2.5 + 408.0335 = 420

410.5335 = 420

Therefore, the distance from city A to city C is approximately 410.53 km.

Answer

The distance from city A to city C is approximately 410.53 km.