Question

Задача.Одна труба может заполнить бассейн на 24 часа быстрее, чем другая. Через 8 часов после включения второй трубы включили первую и через 20 часов совместной работы обеих труб оказалось, что заполнено 2\3 бассейна. За какое время может заполнить бассейн каждая труба, работая самостоятельно?

Answer 1

Примем весь бассейн(всю работу) за 1.
х ч работает 1я труба, заполняя весь бассейн
х+24 -2я труба
1/х производительность 1й трубы
1/(х+24) производительность 2й трубы
20+8 =28 часов работала 2я труба, а 1я- 20ч
20/х+28/(х+24)=2/3
3*20(х+24)+3х*28=2х(х+24)
60х+1440+84х=2х²+48х
2х²-96х-1440=0
х²-48х-720=0
Д=48²+4*720=2304+2880=5184=72²
х₁=(48-72)/2=-24:2=-12 не удовл условию
х₂=(48+72)/2=120:2=60 ч 1я труба
60+24=84 ч - 2я труба
ответ: первая за 60 часов; вторая за 84 часа

Answer 2

Problem Analysis

We are given two pipes that can fill a pool at different rates. The first pipe can fill the pool on its own in 24 hours less than the second pipe. After 8 hours of the second pipe being turned on, the first pipe is also turned on. After 20 hours of both pipes working together, 2/3 of the pool is filled. We need to determine how long it would take each pipe to fill the pool on its own.

Solution

Let's assume that the first pipe can fill the pool in x hours and the second pipe can fill the pool in y hours.

According to the problem, the first pipe can fill the pool in 24 hours less than the second pipe. So, we can write the equation:

x = y - 24 After 8 hours of the second pipe being turned on, the first pipe is also turned on. So, the second pipe has been working for 8 hours before the first pipe starts. Therefore, the amount of work done by the second pipe in 8 hours is:

8/y After 20 hours of both pipes working together, 2/3 of the pool is filled. So, the amount of work done by both pipes in 20 hours is:

20(1/x + 1/y) According to the problem, 2/3 of the pool is filled after 20 hours. So, we can write the equation:

20(1/x + 1/y) = 2/3 Now, we have two equations (equations and with two variables (x and y). We can solve these equations to find the values of x and y.

Solving the Equations

Let's solve equations and to find the values of x and y.

From equation we have:

x = y - 24

Substituting this value of x into equation we get:

20(1/(y - 24) + 1/y) = 2/3

Simplifying the equation, we get:

20(y + y - 24)/(y(y - 24)) = 2/3

Cross-multiplying, we get:

60(y + y - 24) = 2(y(y - 24))

Expanding and simplifying, we get:

120y - 2880 = 2y^2 - 48y

Rearranging the equation, we get:

2y^2 - 168y + 2880 = 0

Solving this quadratic equation, we find two possible values for y:

y = 60 or y = 24

Substituting these values of y back into equation we can find the corresponding values of x:

For y = 60:

x = 60 - 24 = 36

For y = 24:

x = 24 - 24 = 0

Since it doesn't make sense for the first pipe to take 0 hours to fill the pool, we can conclude that the first pipe takes 36 hours to fill the pool on its own, while the second pipe takes 60 hours to fill the pool on its own.

Answer

Therefore, the first pipe can fill the pool on its own in 36 hours, and the second pipe can fill the pool on its own in 60 hours.