Question

Первому рабочему для выполнения задания надо на 4 ч меньше, чем второму. Первый рабочий проработал 2 ч, а затем его сменил второй. После того как второй рабочий проработал 3 ч, оказалось, что выполнено 1 дробь 2 задания. За сколько часов может выполнить это задание каждый рабочий самостоятельно?

Answer 1

Объем работы  - 1
I рабочий :
Время  на выполнение работы- х
Производительность  -  (1/х) 
II рабочий:
Время  - (х+4)
Производительность  -  1/ (х+4)
Уравнение:
2 *  (1/х) + 3 * (1/(х+4) = 1/2
2/х   +  3/(х+4) = 1/2
2*2 *(х+4) + 3*2*х = 1  * х *(х+4)
4х+16 + 6х=х²+4х
10х+16-х²-4х= 0
-х²+6х+16=0
D= 36-4*(-1)*16= 36+64=100
D>0  - два корня
х₁= (-6+10)  / (2*(-1)) = -2  - не удовлетворяет условию задачи
х₂= (-6-10)/2 *(-1) = (-16) / (-2) = 8  часов -   I рабочий
8+4 = 12 часов  - II рабочий

Ответ:  за 8 часов выполнит задание I рабочий самостоятельно ,  
за 12 часов  - II рабочий .

Answer 2

Task Analysis

To solve this task, we need to determine the amount of time it would take for each worker to complete the task individually. We are given the following information:

- The first worker needs 4 hours less than the second worker to complete the task. - The first worker worked for 2 hours before being replaced by the second worker. - After the second worker worked for 3 hours, it was found that only 1/2 of the task was completed.

We need to find the time it would take for each worker to complete the task individually.

Solution

Let's assume that the second worker takes x hours to complete the task. Since the first worker needs 4 hours less than the second worker, the first worker would take (x - 4) hours to complete the task.

According to the given information, the first worker worked for 2 hours and then was replaced by the second worker. After the second worker worked for 3 hours, only 1/2 of the task was completed. This means that the second worker completed 1/2 of the task in 3 hours.

To find the time it would take for each worker to complete the task individually, we can set up the following equation:

2/(x - 4) + 3/x = 1/2

Let's solve this equation to find the value of x.

Equation Solution

To solve the equation, we can start by multiplying both sides of the equation by 2x(x - 4) to eliminate the denominators:

2x + 6(x - 4) = x(x - 4)/2

Simplifying the equation:

2x + 6x - 24 = (x^2 - 4x)/2

Combining like terms:

8x - 24 = (x^2 - 4x)/2

Multiplying both sides of the equation by 2 to eliminate the fraction:

16x - 48 = x^2 - 4x

Rearranging the equation:

x^2 - 20x + 48 = 0

Now we have a quadratic equation. We can solve it by factoring or using the quadratic formula. Let's use the quadratic formula to find the value of x.

Quadratic Formula

The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation x^2 - 20x + 48 = 0, the coefficients are:

a = 1, b = -20, c = 48

Substituting these values into the quadratic formula:

x = (-(-20) ± √((-20)^2 - 4(1)(48))) / (2(1))

Simplifying:

x = (20 ± √(400 - 192)) / 2

x = (20 ± √208) / 2

x = (20 ± 4√13) / 2

Simplifying further:

x = 10 ± 2√13

So, the possible values for x are:

x = 10 + 2√13 x = 10 - 2√13

These are the two possible times it would take for the second worker to complete the task individually. To find the time it would take for the first worker, we subtract 4 hours from each of these values:

For the first worker: - Time = x - 4

Let's calculate the values for both workers.

Calculation

For the first worker: - Time = 10 + 2√13 - 4 = 6 + 2√13

For the second worker: - Time = 10 - 2√13 - 4 = 6 - 2√13

Therefore, the first worker can complete the task individually in 6 + 2√13 hours, and the second worker can complete the task individually in 6 - 2√13 hours.

Please note that the values of x and the resulting times are approximate, as they involve the square root of 13, which is an irrational number.

Let me know if there's anything else I can help you with!