Question

Двое рабочих выполнили некоторую работу за 8 ч. Первый из них, работая отдельно, может выполнить ту же работу на 12 ч быстрее второго, если тот будет работать отдельно. За сколько часов второй рабочий один может выполнить ту же работу?​

Answer 1

Ответ:

Хч - нужно 1 рабочему

(х+12) ч - нужно 2 рабочему

1/х - производительность 1 рабочего

1/(х+12) - производительность 2 рабочего

1/х + 1/(х+12) =1/8

8(х+12)+8х=х(х+12)

8х+96+8х=х^2+12х

Х^2-4х-96=0

D=(-4)^2-4×1×(-96)=16+384=400

Х1=(4-20)/2= - 8 не подходит

Х2=(4+20)/2=12 ч нужно 1 рабочему

12+12=24 ч нужно 2 рабочему

Ответ : 24 ч

Answer 2

Problem Analysis

We have two workers who completed a certain amount of work in 8 hours. The first worker can complete the same work 12 hours faster than the second worker if they work separately. We need to determine how many hours the second worker can complete the same work alone.

Solution

Let's assume that the second worker can complete the work alone in x hours.

According to the given information, the first worker can complete the work in x + 12 hours if they work separately.

We know that the first worker completed the work in 8 hours, so we can set up the following equation:

1/8 + 1/(x + 12) = 1/x

To solve this equation, we can find a common denominator and simplify:

(x + 12 + 8) / (8(x + 12)) = 1/x

(x + 20) / (8x + 96) = 1/x

Cross-multiplying, we get:

x(x + 20) = 8x + 96

Expanding and rearranging the equation:

x^2 + 20x = 8x + 96

x^2 + 20x - 8x - 96 = 0

x^2 + 12x - 96 = 0

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

Quadratic Equation Solution

Using the quadratic formula, we have:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation x^2 + 12x - 96 = 0, the coefficients are: a = 1, b = 12, c = -96

Substituting these values into the quadratic formula:

x = (-12 ± √(12^2 - 4(1)(-96))) / (2(1))

Simplifying further:

x = (-12 ± √(144 + 384)) / 2

x = (-12 ± √528) / 2

x = (-12 ± 23.02) / 2

We have two possible solutions for x:

x1 = (-12 + 23.02) / 2 = 11.02 / 2 = 5.51

x2 = (-12 - 23.02) / 2 = -35.02 / 2 = -17.51

Since time cannot be negative, we discard the negative solution.

Answer

Therefore, the second worker can complete the same work alone in approximately 5.51 hours.