Question

двое рабочих могут выполнить некоторую работу за 7 дней при условии, что второй приступит к ней на 2 дня позже первого. если бы ту же работу каждый выполнял в одиночку, то первому потребовалось бы на 4 дня больше, чем второму. за сколько дней каждый рабочий мог бы выполнить эту работу?

Answer 1

Такие задачи удобно решать составляя таблицу

                              V             t                     A
1 рабочий           1/(х+4)    х+4 дней         1
2 рабочий           1/х           x дней             1

1+2 рабочий   1/(х+4)+1/х    2 дня 1 раб   1
                                              5 дней вместе

составим уравнение

 $\displaystyle 2( \frac{1}{x+4})+5( \frac{1}{x+4}+ \frac{1}{x})=1\\\\ \frac{2}{x+4}+5( \frac{2x+4}{x(x+4)})=1\\\\ \frac{2x}{x(x+4)}+ \frac{10x+20}{x(x+4)}= \frac{x(x+4)}{x(x+4)}\\\\2x+10x+20=x^2+4x\\\\x^2-8x-20=0\\\\D=64+80=144=12^2\\\\x_{1.2}= \frac{8\pm 12}{2}\\\\x_1=10; x_2=-2$

Значит время второго  10 дней
время первого 10+4=14 дней  

Answer 2

Problem Analysis

We are given that two workers can complete a certain job in 7 days, with the second worker starting 2 days after the first worker. If each worker were to work alone, the first worker would take 4 days longer than the second worker to complete the job. We need to determine how many days each worker would take to complete the job if they worked alone.

Solution

Let's assume that the first worker can complete the job in x days when working alone. According to the given information, the second worker would take x + 4 days to complete the job alone.

To find the solution, we can set up the following equation based on the work rates of the two workers:

1/x + 1/(x+4) = 1/7

Simplifying this equation will allow us to solve for x, which represents the number of days the first worker would take to complete the job alone.

Calculation

To solve the equation, we can multiply both sides by 7x(x+4) to eliminate the denominators:

7(x+4) + 7x = x(x+4)

Expanding and simplifying the equation:

7x + 28 + 7x = x^2 + 4x

14x + 28 = x^2 + 4x

Rearranging the equation:

x^2 - 10x - 28 = 0

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

Using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = -10, and c = -28.

Substituting these values into the quadratic formula:

x = (-(-10) ± √((-10)^2 - 4(1)(-28))) / (2(1))

Simplifying further:

x = (10 ± √(100 + 112)) / 2

x = (10 ± √212) / 2

Calculating the square root of 212:

√212 ≈ 14.56

Now we can substitute this value into the equation:

x = (10 ± 14.56) / 2

Calculating both possibilities:

x1 = (10 + 14.56) / 2 ≈ 12.28 x2 = (10 - 14.56) / 2 ≈ -2.28

Since we are dealing with time, we can discard the negative value. Therefore, the first worker would take approximately 12.28 days to complete the job alone.

To find the number of days the second worker would take to complete the job alone, we can add 4 to the value of x1:

x2 = x1 + 4 ≈ 12.28 + 4 ≈ 16.28

Therefore, the second worker would take approximately 16.28 days to complete the job alone.

Answer

If each worker were to work alone, the first worker would take approximately 12.28 days to complete the job, and the second worker would take approximately 16.28 days to complete the job.