Question

Бассейн наполняется двумя трубами,действующими одновременно,за 4 часа.За сколько часов может наполнить бассейн первая труба, действуя в отдельности, если она наполняет бассейн на 6 ч. дольше, чем вторая? (НУЖНО РЕШИТЬ СИСТЕМОЙ УРАВНЕНИЙ)

Answer 1

4/х+4/у=1
х-у=6
ответ 12 

Answer 2

Problem Analysis

We are given that a pool is being filled by two pipes working simultaneously, and it takes 4 hours to fill the pool. We need to determine how long it would take for the first pipe to fill the pool on its own if it takes 6 hours longer than the second pipe.

Solution

Let's assume that the second pipe takes x hours to fill the pool on its own. According to the problem, the first pipe takes 6 hours longer than the second pipe, so it would take (x + 6) hours to fill the pool on its own.

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

1/((x + 6)) + 1/x = 1/4

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

Calculation

To solve the equation, we can multiply both sides by 4x(x + 6) to eliminate the denominators:

4x + 4(x + 6) = x(x + 6)

Expanding and simplifying the equation:

4x + 4x + 24 = x^2 + 6x

8x + 24 = x^2 + 6x

Rearranging the equation:

x^2 - 2x - 24 = 0

Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = -2, and c = -24. Substituting these values into the quadratic formula:

x = (-(-2) ± √((-2)^2 - 4(1)(-24))) / (2(1))

Simplifying:

x = (2 ± √(4 + 96)) / 2

x = (2 ± √100) / 2

x = (2 ± 10) / 2

We have two possible solutions:

1. x = (2 + 10) / 2 = 12 / 2 = 6 2. x = (2 - 10) / 2 = -8 / 2 = -4

Since the time cannot be negative, we discard the second solution.

Therefore, the second pipe takes 6 hours to fill the pool on its own, and the first pipe takes 6 hours longer, which is 12 hours.

Answer

The first pipe, working on its own, would take 12 hours to fill the pool.

Verification

To verify our answer, let's check if it satisfies the original equation:

1/((12 + 6)) + 1/12 = 1/4

Simplifying:

1/18 + 1/12 = 1/4

Finding a common denominator:

2/36 + 3/36 = 9/36

Simplifying:

5/36 = 9/36

The equation is true, which confirms that our answer is correct.