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 20 км/ч, а вторую половину пути-со скоростью 120км/ч, в результате чего прибыл в В одновременно с первым автомобилистом. Найдите скорость первого автомобилиста, если известно, что она больше 70км/ч. Ответ дайте в км/ч

Answer 1

решается по определению средней скорости

Answer 2

Problem Analysis

We are given that two drivers, A and B, simultaneously traveled from point A 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 20 km/h slower than driver A and the second half at a speed of 120 km/h. We need to find the speed of driver A, given that it is greater than 70 km/h.

Solution

Let's assume the total distance from A to B is d km.

We know that driver B traveled the first half of the distance at a speed 20 km/h slower than driver A. Let's denote the speed of driver A as x km/h. Therefore, the speed of driver B for the first half of the distance is (x - 20) km/h.

We also know that driver B traveled the second half of the distance at a speed of 120 km/h.

To find the time taken by driver A and driver B, we can use the formula:

Time = Distance / Speed

For driver A: - Time taken by driver A = Total distance / Speed of driver A - Time taken by driver A = d / x

For driver B: - Time taken by driver B for the first half of the distance = First half distance / Speed of driver B for the first half of the distance - Time taken by driver B for the first half of the distance = (d / 2) / (x - 20)

- Time taken by driver B for the second half of the distance = Second half distance / Speed of driver B for the second half of the distance - Time taken by driver B for the second half of the distance = (d / 2) / 120

Since both drivers arrived at point B simultaneously, the total time taken by driver A and driver B should be the same:

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

Substituting the values, we get:

d / x = (d / 2) / (x - 20) + (d / 2) / 120

To solve this equation for x, we can multiply both sides by x(x - 20)(120) to eliminate the denominators:

d * (x - 20)(120) = (d / 2) * x * (120) + (d / 2) * (x - 20) * x

Simplifying the equation:

d * (x - 20)(120) = (d / 2) * (120x + x^2 - 20x)

d * (x - 20)(120) = (d / 2) * (x^2 + 100x)

2 * d * (x - 20)(120) = d * (x^2 + 100x)

240 * (x - 20) = x^2 + 100x

240x - 4800 = x^2 + 100x

x^2 - 140x + 4800 = 0

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

Using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a = 1, b = -140, and c = 4800.

Substituting the values, we get:

x = (-(-140) ± √((-140)^2 - 4 * 1 * 4800)) / (2 * 1)

x = (140 ± √(19600 - 19200)) / 2

x = (140 ± √400) / 2

x = (140 ± 20) / 2

Simplifying further:

x = (140 + 20) / 2 = 160 / 2 = 80

or

x = (140 - 20) / 2 = 120 / 2 = 60

Since we are given that the speed of driver A is greater than 70 km/h, the solution is:

The speed of the first driver (driver A) is 80 km/h.

Answer

The speed of the first driver (driver A) is 80 km/h.