Question

Помогите, пожалуйста, с подробным решением!!! Из города A в город B ведут две дороги: шоссе, длина которого 75 км, и просёлочная дорога, длина которой 81 км. Из города A в город B одновременно выехали автобус и грузовая машина. Автобус ехал по шоссе и проезжал в час на 5 км больше, чем грузвая машина, которая ехала по просёлочной дороге. Автобус прибыл в город B на 18 минут раньше грузовой машины. Найди скорость грузовой машины.

Answer 1

Х км/ч - скорость гр.машины
(х+5) км/ч- скорость автобуса
18мин.=18:60=0,3ч.
81/х - 75/(х+5)=0,3
81(х+5)-75х=0,3х(х+5)
6х+405-0,3х²-1,5х=0
0,3х²-4,5х-405=0  I :0,3
х²-15х-1350=0
D=b²-4ac=(-15)²-4·(-1350)=225+5400=5625
$x_{1} =(15+75)/2=45$ км/ч - скорость гр. машины
$x_{2} =(15-75)/2=-30$ - не подходит

Answer 2

Problem Analysis

We are given that there are two roads leading from city A to city B: a highway that is 75 km long and a dirt road that is 81 km long. A bus and a truck left city A for city B at the same time. The bus traveled on the highway and drove 5 km per hour faster than the truck, which traveled on the dirt road. The bus arrived in city B 18 minutes earlier than the truck. We need to find the speed of the truck.

Solution

Let's assume the speed of the truck is x km/h. Since the bus traveled 5 km/h faster than the truck, the speed of the bus is x + 5 km/h.

To find the time it took for each vehicle to travel their respective distances, we can use the formula time = distance / speed.

The time it took for the bus to travel the distance of the highway (75 km) is 75 / (x + 5) hours.

The time it took for the truck to travel the distance of the dirt road (81 km) is 81 / x hours.

We are given that the bus arrived in city B 18 minutes earlier than the truck. Since 18 minutes is equal to 18/60 = 0.3 hours, we can set up the following equation:

(75 / (x + 5)) - (81 / x) = 0.3

To solve this equation, we can multiply through by x(x + 5) to eliminate the denominators:

75x - 81(x + 5) = 0.3x(x + 5)

Simplifying the equation:

75x - 81x - 405 = 0.3x^2 + 1.5x

0.3x^2 + 1.5x - 6x - 405 = 0

0.3x^2 - 4.5x - 405 = 0

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

Quadratic Equation Solution

Using the quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a), where a = 0.3, b = -4.5, and c = -405, we can find the two possible values of x.

Calculating the discriminant, D = b^2 - 4ac:

D = (-4.5)^2 - 4(0.3)(-405) = 20.25 + 486 = 506.25

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

Using the quadratic formula:

x = (-(-4.5) ± √(506.25)) / (2(0.3))

Simplifying:

x = (4.5 ± √(506.25)) / 0.6

Calculating the square root of 506.25:

√(506.25) ≈ 22.5

Substituting the values:

x = (4.5 ± 22.5) / 0.6

Calculating the two possible values of x:

x₁ = (4.5 + 22.5) / 0.6 ≈ 45 km/h

x₂ = (4.5 - 22.5) / 0.6 ≈ -30 km/h

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

Answer

The speed of the truck is approximately 45 km/h.