Question

Перша труба може заповнити басейн 24 години швидче,ніж друга.Спочатку відкрили другу трубу,а через 8 годин - першу. Через 20 годин спільної двох труб водою було заповнений 2/3 басейну.За скільки годин може заповнити басейн кожна труба працюючи самостійно?

Answer 1

Решение дано на фото.

Answer 2

Problem Analysis

We are given that the first pipe can fill a pool 24 hours faster than the second pipe. The second pipe was opened first, and after 8 hours, the first pipe was also opened. After a total of 20 hours, the pool was filled to 2/3 of its capacity. We need to determine how long each pipe would take to fill the pool if they were working independently.

Solution

Let's assume that the second pipe takes x hours to fill the pool on its own. Since the first pipe is 24 hours faster, it would take (x - 24) hours to fill the pool on its own.

We are given that after 8 hours, the first pipe was opened. This means that in those 8 hours, the second pipe filled 8/x of the pool's capacity. Therefore, after 8 hours, the remaining capacity to be filled is 1 - (8/x).

After 20 hours, the combined work of both pipes filled 2/3 of the pool's capacity. This means that in those 20 hours, the combined work of both pipes filled 2/3 - (1 - (8/x)) of the pool's capacity.

To find the individual rates of each pipe, we can set up the following equation:

(20 / x) + (20 / (x - 24)) = 2/3 - (1 - (8/x))

Simplifying this equation will give us the value of x, which represents the number of hours it takes for the second pipe to fill the pool on its own.

Let's solve this equation step by step:

(20 / x) + (20 / (x - 24)) = 2/3 - (1 - (8/x))

Multiplying through by 3x(x - 24) to eliminate the denominators:

60(x - 24) + 60x = 2x(x - 24) - 3(x - 24)(8/x)

Expanding and simplifying:

60x - 1440 + 60x = 2x^2 - 48x - 24(8)

120x - 1440 = 2x^2 - 48x - 192

Rearranging to form a quadratic equation:

2x^2 - 168x + 648 = 0

Solving this quadratic equation will give us the value of x, which represents the number of hours it takes for the second pipe to fill the pool on its own.

Using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 2, b = -168, and c = 648.

Let's calculate the value of x:

x = (-(-168) ± √((-168)^2 - 4(2)(648))) / (2(2))

x = (168 ± √(28224 - 5184)) / 4

x = (168 ± √(23040)) / 4

x = (168 ± 480) / 4

x = (168 + 480) / 4 or x = (168 - 480) / 4

x = 648 / 4 or x = -312 / 4

x = 162 or x = -78

Since the number of hours cannot be negative, we can conclude that the second pipe takes 162 hours to fill the pool on its own.

To find the time it takes for the first pipe to fill the pool on its own, we subtract 24 hours from the time taken by the second pipe:

Time taken by the first pipe = 162 - 24 = 138 hours.

Therefore, the first pipe takes 138 hours to fill the pool on its own, and the second pipe takes 162 hours to fill the pool on its own.

Answer

The first pipe takes 138 hours to fill the pool on its own, and the second pipe takes 162 hours to fill the pool on its own.