Question

Два токаря ,работая вместе,могут выполнить заказ за 7 дней,причем второй начинает работу на 1,5 дня позже первого.За сколько дней каждый из них может выполнить этот заказ,работая отдельно, если второму потребуется на 3 дня меньше,чем первому?

Answer 1

Эту задачу можно "расколоть" с помощью уравнения. Составить его можно так. Пусть 1й выполнит весь заказ за x дней, тогда 2й за x-3 дней. Если принять весь объём работ за 1, то скорость работы 1-го будет:
$\frac{1}{x}$
а скорость работы 2-го:
$\frac{1}{x-3}$
Если они будут выполнять заказ совместно так, как указано в условии, то за 7 дней они выполнят часть работы:
$7\cdot\frac{1}{x}+(7-1,5)\cdot\frac{1}{x-3}$
Что по условию равно всему объёму работ, т. е. 1. Итак мы получаем уравнение:
$\frac{7}{x}+\frac{5,5}{x-3}=1$
Решаем его:
$\frac{7(x-3)+5,5x}{x(x-3)}=1 \\ \\ x \neq 0; \\ x \neq 3; \\ x \ \textgreater \ 0 ; \\ 7x-21+5,5x=x^2-3x \\ \\ x^2-15,5x+21=0 \\ 10x^2-155x+210=0 \\ D=155^2-4*10*210=15625 \\ \\ x_{1,2}= \frac{155 \pm 125}{20} \\ x_{1}=14 \\ x_{2}= \frac{3}{2}=1,5$

При x=1,5 2й должен выполнить заказ за 1,5-3=-1,5 дня, а так не бывает.
Остаётся вариант x=14. Тогда 2й выполнит заказ за 14-3=11 дней.

Ответ: 1й может выполнить заказ за 14 дней, 2й за 11 дней

Answer 2

Problem Analysis

We have two lathe workers who can complete an order in 7 days. The second worker starts working 1.5 days later than the first worker. The second worker takes 3 days less than the first worker to complete the order. We need to determine how many days each worker would take to complete the order if they worked separately.

Solution

Let's assume that the first worker takes x days to complete the order when working alone. Since the second worker starts 1.5 days later, the first worker would have already completed some portion of the order by the time the second worker starts. Let's calculate that portion.

The first worker completes 1/x of the order per day. In 1.5 days, the first worker would have completed (1.5/x) of the order.

Since the second worker takes 3 days less than the first worker to complete the order, the second worker would take (x - 3) days to complete the remaining portion of the order.

The total time taken by the second worker to complete the order is (1.5/x) + (x - 3) days.

According to the problem, the total time taken by both workers working together is 7 days. Therefore, we can write the equation:

(1/x) + (1.5/x) + (x - 3) = 1/7

Simplifying the equation, we get:

(2.5/x) + (x - 3) = 1/7

Multiplying both sides of the equation by 7x, we get:

17.5 + 7x^2 - 21x = x

Rearranging the equation, we get:

7x^2 - 22x + 17.5 = 0

Solving this quadratic equation will give us the value of x, which represents the number of days the first worker takes to complete the order when working alone.

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

Calculation

Using the quadratic formula, we can find the value of x:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 7, b = -22, and c = 17.5.

Plugging in the values, we get:

x = (-(-22) ± √((-22)^2 - 4 * 7 * 17.5)) / (2 * 7)

Simplifying further:

x = (22 ± √(484 - 490)) / 14

x = (22 ± √(-6)) / 14

Since the square root of a negative number is not a real number, there are no real solutions to this equation. Therefore, we cannot determine the number of days the first worker takes to complete the order when working alone.

Conclusion

Based on the given information, we cannot determine the number of days each worker would take to complete the order when working separately.