Question

Двое рабочих выполняют некоторую работу.После 45 минут совместной работы первый рабочий был переведён на другую работу, и второй рабочий закончил оставшуюся часть работы за 2 часа 15 минут.За какое время мог бы выполнить всю работу каждый рабочий в отдельности,если второму на это понадобится на 1 час больше,чем первому?

Answer 1

Пусть время работы первого рабочего х часов     (Х строго больше нуля)    , тогда второго - ( х +1 )часов. Производительность труда первого 1\х, второго - 1\ (х+1).⇒ 
(1\х + 1\ (х+1) ) * 3\4 + 1\ (х+1) * 2 1\4 = 1
3 \ 4х +3 \ 4(х+1) +9 \ 4 (х+1) = 1
3(х+1) + 3х + 9х = 4х² +4х
4х² - 11х - 3 =0 ⇒ Д = 11² - 4*4*(-3) = 169 ⇒ √Д= 13 
х₁= (11+13) \ 2*4 = 3   х₂=(11-13) \ 2*4 = - 1\4 - ∉по условию задачи
значит первый рабочий самостоятельно мог бы выполнить данную работу за 3 часа, а второй за 3+1= 4 часа

Answer 2

Problem Analysis

We have two workers who initially work together for 45 minutes. After that, the first worker is transferred to another task, and the second worker completes the remaining work in 2 hours and 15 minutes. We need to determine how long it would take each worker to complete the entire task individually, with the second worker taking 1 hour longer than the first.

Solution

Let's assume that the first worker can complete the task in x hours. Since the second worker takes 1 hour longer, they would take (x + 1) hours to complete the task.

From the given information, we know that the second worker completes the remaining work in 2 hours and 15 minutes, which is equivalent to 2.25 hours.

To solve this problem, we can set up the following equation based on the work rates of the two workers:

1/x + 1/(x + 1) = 1/2.25

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

Calculation

To solve the equation, we can multiply all terms by the least common denominator (LCD) of x(x + 1)(2.25), which is 2.25x(x + 1).

2.25(x + 1) + 2.25x = x(x + 1)

Expanding and simplifying the equation:

2.25x + 2.25 + 2.25x = x^2 + x

4.5x + 2.25 = x^2 + x

Rearranging the equation:

x^2 + x - 4.5x - 2.25 = 0

x^2 - 3.5x - 2.25 = 0

Now, we can solve this quadratic equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = -3.5, and c = -2.25.

Substituting the values into the quadratic formula:

x = (-(-3.5) ± √((-3.5)^2 - 4(1)(-2.25))) / (2(1))

Simplifying further:

x = (3.5 ± √(12.25 + 9)) / 2

x = (3.5 ± √21.25) / 2

Calculating the square root:

x = (3.5 ± 4.61) / 2

Now, we have two possible solutions:

x1 = (3.5 + 4.61) / 2 ≈ 4.055

x2 = (3.5 - 4.61) / 2 ≈ -0.555 (not a valid solution since time cannot be negative)

Therefore, the first worker can complete the entire task in approximately 4.055 hours.

Since the second worker takes 1 hour longer, they would take approximately 5.055 hours to complete the entire task.

Answer

The first worker can complete the entire task in approximately 4.055 hours, while the second worker would take approximately 5.055 hours to complete the entire task.