Question

Из пункта A в пункт B одновременно выехали два автомобилиста. Первый проехал с постоянной скоростью весь путь. Второй проехал первую половину пути со скоростью, меньшей скорости первого на 11 км/ч, а вторую половину пути - со скоростью 66 км/ч, в результате чего прибыл в пункт B одновременно с первым автомобилистом. Найдите скорость первого автомобилиста, если известно, что она больше 40 км/ч. Ответ дайте в километрах в час

Answer 1

Обозначим расстояние от А до В  равна S, скорость первого x. Тогда
первый ехал S/x часов, а второй $\frac{S}{2(x-11)} + \frac{S}{2*66}$
получаем уравнение
 $\frac{S}{x} =\frac{S}{2(x-11)} + \frac{S}{2*66} \\ \frac{1}{x} =\frac{1}{2(x-11)} + \frac{1}{2*66} \\ \frac{2}{x} =\frac{1}{(x-11)} + \frac{1}{66}$

$\frac{2}{x} =\frac{1}{(x-11)} + \frac{1}{66} \\ \frac{2}{x} =\frac{66+(x-11)}{66(x-11)} \\ \frac{2}{x}=\frac{55+x}{66(x-11)}$

2*66(x-11)=x(55+x)
132x-1452=55x+x²
x²-77x+1452=0
D=77²-4*1452=121
√D=11
x₁=(77-11)/2=33<40 отбрасываем
x₂=(77+11)/2=44
Ответ: 44 км/ч

Answer 2

Problem Analysis

We are given that two drivers simultaneously traveled from point A to point B. The first driver traveled the entire distance at a constant speed. The second driver traveled the first half of the distance at a speed 11 km/h slower than the first driver and the second half of the distance at a speed of 66 km/h. We need to find the speed of the first driver, given that it is greater than 40 km/h.

Solution

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

The first driver traveled the entire distance at a constant speed, which we'll call V1 km/h.

The second driver traveled the first half of the distance at a speed 11 km/h slower than the first driver, so their speed was (V1 - 11) km/h. The second half of the distance was traveled at a speed of 66 km/h.

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

Time = Distance / Speed

For the first driver: Time taken by the first driver = D / V1

For the second driver: Time taken for the first half of the distance = (D/2) / (V1 - 11) Time taken for the second half of the distance = (D/2) / 66

Since both drivers arrived at point B at the same time, we can equate their times:

D / V1 = (D/2) / (V1 - 11) + (D/2) / 66

To solve this equation for V1, we can multiply both sides by V1(V1 - 11)(66):

D * (V1 - 11)(66) = (D/2) * V1 * (V1 - 11) + (D/2) * (V1)(66)

Simplifying the equation:

66D(V1 - 11) = (D/2)(V1^2 - 11V1) + 33D(V1)

Now, let's solve the equation to find the value of V1.

Calculation

Let's substitute D = 1 (since the actual value of D doesn't matter for finding V1).

66(1)(V1 - 11) = (1/2)(V1^2 - 11V1) + 33(1)(V1)

66V1 - 726 = (1/2)(V1^2 - 11V1) + 33V1

Multiplying both sides by 2 to eliminate the fraction:

132V1 - 1452 = V1^2 - 11V1 + 66V1

Rearranging the equation:

V1^2 - 77V1 + 1452 = 0

Using the quadratic formula:

V1 = (-b ± √(b^2 - 4ac)) / 2a

where a = 1, b = -77, and c = 1452.

Calculating the discriminant:

√(b^2 - 4ac) = √((-77)^2 - 4(1)(1452)) = √(5929 - 5808) = √121 = 11

Using the quadratic formula:

V1 = (-(-77) ± 11) / 2(1) = (77 ± 11) / 2 = (88 / 2) or (66 / 2)

V1 = 44 or 33

Since we know that V1 is greater than 40 km/h, the speed of the first driver is 44 km/h.

Answer

The speed of the first driver is 44 km/h.