Question

Допоможіть!!! Якщо одночасно відкрити дві труби різної пропускної спроможності, то басейн буде заповнено за 6 год. Якщо відкрити обидві труби лише на 2 год, а потім залишити відкритою тільки одну з них, то вона заповнить решту басейна за 10 год. За скільки годин кожна труба може самостійно наповнити басейн?​

Answer 1

Problem Analysis

We are given information about filling a pool using two pipes of different capacities. By analyzing the given information, we can determine the time it takes for each pipe to fill the pool individually.

Given Information

Let's summarize the given information: - If both pipes are opened simultaneously, the pool is filled in 6 hours. - If both pipes are opened for 2 hours and then one pipe is closed, the remaining pipe fills the rest of the pool in 10 hours.

Solution

Let's assume that the first pipe fills the pool at a rate of x units per hour, and the second pipe fills the pool at a rate of y units per hour.

From the given information, we can create the following equations:

1. If both pipes are opened simultaneously, the pool is filled in 6 hours: - The total capacity of the pool is filled in 6 hours: 6(x + y).

2. If both pipes are opened for 2 hours and then one pipe is closed, the remaining pipe fills the rest of the pool in 10 hours: - In the first 2 hours, both pipes fill a total of 2(x + y) units of the pool's capacity. - The remaining capacity of the pool is filled by one pipe in 10 hours: 10(x or y).

Now, we can set up an equation using the above information to solve for the individual rates of each pipe.

Solving the Equations

Using the equations derived from the given information, we can solve for the individual rates of each pipe.

Equation 1: 6(x + y) = pool capacity (1) Equation 2: 2(x + y) + 10(x or y) = pool capacity (2)

To solve these equations, we need to find the value of x or y.

From Equation 2, we can rewrite it as: 2(x + y) + 10x = pool capacity (3)

Simplifying Equation 3, we get: 2x + 2y + 10x = pool capacity 12x + 2y = pool capacity (4)

Substituting Equation 1 into Equation 4, we get: 12x + 2y = 6(x + y) 12x + 2y = 6x + 6y 6x = 4y x = (4/6)y x = (2/3)y (5)

Now, we can substitute the value of x from Equation 5 into Equation 1 to find the value of y.

Substituting x = (2/3)y into Equation 1, we get: 6((2/3)y + y) = pool capacity 6((2/3)y + (3/3)y) = pool capacity 6((5/3)y) = pool capacity 10y = pool capacity y = (1/10)pool capacity (6)

Finally, we can substitute the value of y from Equation 6 into Equation 5 to find the value of x.

Substituting y = (1/10)pool capacity into Equation 5, we get: x = (2/3)(1/10)pool capacity x = (1/15)pool capacity (7)

Conclusion

From the analysis and calculations, we have determined the individual rates of each pipe: - The first pipe can fill the pool at a rate of (1/15)pool capacity per hour. - The second pipe can fill the pool at a rate of (1/10)pool capacity per hour.

Please note that the actual values of the pool capacity and the rates of the pipes are not provided in the given information, so we cannot determine the exact time it takes for each pipe to fill the pool. However, we have derived the relationships between the rates of the pipes based on the given information.

If you have any further questions, please let me know!