Question

Двигаясь с постоянной скоростью, автобус проехал 120 км. Если бы он ехал со скоростью на 12км/ч большей, то в дороге находился на 20 минут дольше. С какой скоростью ехал автобус?

Answer 1

Пусть он ехал со скоростью х км/ч. 120 км проехал за 120/х ч. Увеличенная скорость (х+12) км/ч, с ней проедет за 120/(х+12) ч, что на 20 мин или 1/3 быстрее.
120/х-120/(х+12) = 1/3
(120х+1440-120х)/(х(х+12)) = 1/3
4320 = х^2+12х
х^2+12х-4320 = 0
D = 144+4*4320 = 144+17280 = 17424 = (132)^2
x1 = 60
x2 = -144 - не подходит по смыслу.
Ответ: 60 км/ч

Answer 2

Problem Analysis

We are given that a bus traveled a distance of 120 km at a constant speed. If the bus had traveled 12 km/h faster, it would have taken 20 minutes longer to complete the journey. We need to find the speed at which the bus was traveling.

Solution

Let's assume the original speed of the bus is v km/h. According to the problem, if the bus had traveled 12 km/h faster, its speed would have been v + 12 km/h.

We can use the formula distance = speed × time to solve this problem.

The time taken to travel 120 km at the original speed v is given by: time1 = distance / speed1 = 120 / v.

The time taken to travel 120 km at the increased speed v + 12 is given by: time2 = distance / speed2 = 120 / (v + 12).

We are also given that the difference in time between the two scenarios is 20 minutes, which is equal to 1/3 of an hour. So, we have: time2 - time1 = 1/3.

Now, we can substitute the values of time1 and time2 into the equation and solve for v.

Calculation

Let's calculate the value of v using the given information.

time1 = 120 / v

time2 = 120 / (v + 12)

time2 - time1 = 1/3

Substituting the values of time1 and time2 into the equation:

120 / (v + 12) - 120 / v = 1/3

To solve this equation, we can multiply through by 3v(v + 12) to eliminate the denominators:

3v(v + 12) * (120 / (v + 12) - 120 / v) = 3v(v + 12) * (1/3)

Simplifying the equation:

3v(v + 12) * (120v - 120(v + 12)) = v(v + 12)

Expanding and simplifying further:

3v(v + 12) * (120v - 120v - 1440) = v^2 + 12v

3v(v + 12) * (-1440) = v^2 + 12v

-4320v - 51840 = v^2 + 12v

Rearranging the equation:

v^2 + 12v + 4320v + 51840 = 0

v^2 + 4332v + 51840 = 0

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

Using the quadratic formula: v = (-b ± sqrt(b^2 - 4ac)) / (2a), where a = 1, b = 4332, and c = 51840.

Calculating the discriminant: b^2 - 4ac = 4332^2 - 4(1)(51840) = 18736624 - 207360 = 18529264.

Taking the square root of the discriminant: sqrt(18529264) = 4302.9.

Substituting the values into the quadratic formula:

v = (-4332 ± 4302.9) / (2 * 1)

Calculating the two possible values of v:

v1 = (-4332 + 4302.9) / 2 = -29.1 / 2 = -14.55

v2 = (-4332 - 4302.9) / 2 = -8634.9 / 2 = -4317.45

Since speed cannot be negative, we can discard the negative value of v.

Therefore, the speed at which the bus was traveling is approximately v = 14.55 km/h.

Answer

The bus was traveling at a speed of approximately 14.55 km/h.