Question

30 балов Машина половину пути ехала со скоростью на 3км/ч быстрее средней скорости, а вторую половину пути со скоростью в полтора раза меньшей средней. Определите среднюю скорость машины. Ответ выразить в км/ч, округлить до целых.

Answer 1

Дано   V1=Vср+3    V2=Vср/1,5    S1=S2=L/2     Vср- ?

Vср=S/t=S/(t1+t2)=S/(S1/V1+S2/V2)=2*V1*V2/(V1+V2)

пусть Vср=X

X*(V1+V2)=2*V1*V2
X*(X+3 +X/1,5)=2*(X+3)*X/1,5
решаем уравнение получаем X=3 км/ч    V1=6 км/ч    V2=2 км/ч

проверка Vср=2*V1*V2/(V1+V2)=2*6*2/8=3 км/ч
Ответ Vср=3 км/ч

Answer 2

Problem Analysis

We are given that a car traveled half of the distance at a speed 3 km/h faster than the average speed, and the second half of the distance at a speed 1.5 times slower than the average speed. We need to determine the average speed of the car.

Solution

Let's assume the average speed of the car is x km/h.

The first half of the distance is traveled at a speed 3 km/h faster than the average speed, so the speed for the first half is (x + 3) km/h.

The second half of the distance is traveled at a speed 1.5 times slower than the average speed, so the speed for the second half is (x / 1.5) km/h.

Since the car traveled half of the distance at each of these speeds, we can calculate the average speed using the formula:

Average Speed = (Total Distance) / (Total Time)

The total distance is the sum of the distances traveled in the first and second halves, which is equal to half the total distance:

Total Distance = (1/2) * Total Distance

The total time is the sum of the times taken to travel the first and second halves:

Total Time = Time for First Half + Time for Second Half

The time for each half can be calculated using the formula:

Time = Distance / Speed

Substituting the given values into the formulas, we can solve for the average speed.

Calculation

Let's calculate the average speed step by step.

1. Let's assume the total distance is d km. 2. The distance traveled in the first half is (1/2) * d km. 3. The distance traveled in the second half is also (1/2) * d km. 4. The speed for the first half is (x + 3) km/h. 5. The speed for the second half is (x / 1.5) km/h. 6. The time for the first half is [(1/2) * d] / (x + 3) hours. 7. The time for the second half is [(1/2) * d] / (x / 1.5) hours. 8. The total time is [(1/2) * d] / (x + 3) + [(1/2) * d] / (x / 1.5) hours. 9. The average speed is (d / [(1/2) * d] / (x + 3) + [(1/2) * d] / (x / 1.5)) km/h.

Now, let's simplify the equation and solve for x.

Solution

The average speed of the car is x km/h.

Calculation

Let's calculate the average speed step by step.

1. Let's assume the total distance is d km. 2. The distance traveled in the first half is (1/2) * d km. 3. The distance traveled in the second half is also (1/2) * d km. 4. The speed for the first half is (x + 3) km/h. 5. The speed for the second half is (x / 1.5) km/h. 6. The time for the first half is [(1/2) * d] / (x + 3) hours. 7. The time for the second half is [(1/2) * d] / (x / 1.5) hours. 8. The total time is [(1/2) * d] / (x + 3) + [(1/2) * d] / (x / 1.5) hours. 9. The average speed is (d / [(1/2) * d] / (x + 3) + [(1/2) * d] / (x / 1.5)) km/h.

Now, let's simplify the equation and solve for x.

Total Distance = (1/2) * Total Distance = (1/2) * d km

Time for First Half = Distance / Speed = [(1/2) * d] / (x + 3) hours

Time for Second Half = Distance / Speed = [(1/2) * d] / (x / 1.5) hours

Total Time = Time for First Half + Time for Second Half = [(1/2) * d] / (x + 3) + [(1/2) * d] / (x / 1.5) hours

Average Speed = Total Distance / Total Time = (d / [(1/2) * d] / (x + 3) + [(1/2) * d] / (x / 1.5)) km/h

To solve for x, we can multiply both the numerator and denominator of the average speed equation by 2(x + 3)(x / 1.5) to eliminate the fractions:

Average Speed = (d * 2(x + 3)(x / 1.5)) / (d + 2(x + 3)(x / 1.5)) km/h

Simplifying further:

Average Speed = (2d(x + 3)(x / 1.5)) / (d + 2(x + 3)(x / 1.5)) km/h

Now, we can solve for x by setting the average speed equal to x and solving the resulting equation:

(2d(x + 3)(x / 1.5)) / (d + 2(x + 3)(x / 1.5)) = x

Simplifying the equation:

(2d(x + 3)(x / 1.5)) = x(d + 2(x + 3)(x / 1.5))

Expanding and simplifying:

2d(x + 3)(x / 1.5) = x(d + 2(x + 3)(x / 1.5))

2d(x + 3)(x / 1.5) = xd + 2x(x + 3)(x / 1.5)

2d(x^2 + 3x) / 1.5 = xd + 2x^2 + 6x

Multiplying both sides by 1.5 to eliminate the fraction:

2d(x^2 + 3x) = 1.5(xd + 2x^2 + 6x)

Expanding and simplifying:

2dx^2 + 6dx = 1.5xd + 3x^2 + 9x

Rearranging the equation:

2dx^2 + 6dx - 1.5xd - 3x^2 - 9x = 0

Combining like terms:

(2d - 1.5d)x^2 + (6d - 1.5d - 3)x = 0

Simplifying further:

(0.5d)x^2 + (4.5d - 3)x = 0

Factoring out x:

x(0.5dx + 4.5d - 3) = 0

Setting each factor equal to zero:

x = 0 or 0.5dx + 4.5d - 3 = 0

If x = 0, it means the car is not moving, which is not possible in this scenario. Therefore, we can ignore this solution.

Solving the second equation for x:

0.5dx + 4.5d - 3 = 0

0.5dx = 3 - 4.5d

x = (3 - 4.5d) / (0.5d)

Simplifying further:

x = 6 - 9d

Therefore, the average speed of the car is 6 - 9d km/h.

Answer

The average speed of the car is 6 - 9d km/h.