Question

Бассейн наполняется через первую трубу на 6 часов позже чем через первую.При совместной работе обеих труб бассейн наполняется за 4 часа.За сколько часов можно наполнить бассейн через каждую из труб? С решением​

Answer 1

Ответ:

Пошаговое

1/x-1/y=6

1/(x+y)=4

решение:

x+y=1/4

x+y=0.25

y=0.25-x

1/x-1/(0.25-x)=6

(1/x-1/(0.25-x))*x=6*x

(8x-1)/(4x-1)=6x

(8x-1)/(4x-1)*(0.25-x)=6x*(0.25-x)

0.25-2x=-6x²+3x/2

6x²-3.5x+0.25=0

D=(-⁷/₂)²-4*6*0.25=6.25

x₁=(√6.25-(-3.5))/(2*6)=¹/₂ (не удовлетворяет условию )

x₂=(-√6.25-(-3.5))/(2*6)=¹/₁₂

1/(¹/₁₂)=12 часов первая труба

12-6=6 часов вторая труба

проверка

1/(1/12)-1/(1/6)=6

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

Answer 2

Problem Analysis

We are given that a pool is being filled through two pipes, with the second pipe starting 6 hours later than the first pipe. When both pipes are working together, the pool is filled in 4 hours. We need to find the time it takes to fill the pool through each pipe individually.

Solution

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

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

1. The first pipe fills the pool in x hours: - Equation 1: 1/x of the pool is filled per hour by the first pipe.

2. The second pipe fills the pool in y hours, starting 6 hours later than the first pipe: - Equation 2: 1/y of the pool is filled per hour by the second pipe.

3. When both pipes are working together, the pool is filled in 4 hours: - Equation 3: 1/4 of the pool is filled per hour when both pipes are working together.

To solve for x and y, we can set up the following system of equations:

Equation 1 + Equation 2 = Equation 3

1/x + 1/y = 1/4

To solve this system of equations, we can use the method of substitution.

Solution Steps

1. Start with Equation 1: 1/x + 1/y = 1/4. 2. Rearrange Equation 1 to solve for y: 1/y = 1/4 - 1/x. 3. Simplify the right side of the equation: 1/y = (x - 4)/(4x). 4. Take the reciprocal of both sides: y = (4x)/(x - 4). 5. Substitute this value of y into Equation 2: 1/y = 1/x - 1/(x - 4). 6. Simplify the right side of the equation: 1/y = (x - (x - 4))/(x(x - 4)). 7. Simplify further: 1/y = 4/(x(x - 4)). 8. Take the reciprocal of both sides: y = (x(x - 4))/4. 9. Substitute this value of y into Equation 1: 1/x + 1/((x(x - 4))/4) = 1/4. 10. Simplify the equation: 1/x + 4/(x(x - 4)) = 1/4. 11. Multiply both sides of the equation by 4x(x - 4) to eliminate the denominators: 4(x - 4) + 16x = x(x - 4). 12. Expand and simplify the equation: 4x - 16 + 16x = x^2 - 4x. 13. Combine like terms: 20x - 16 = x^2 - 4x. 14. Rearrange the equation to form a quadratic equation: x^2 - 24x + 16 = 0. 15. Solve the quadratic equation using the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a), where a = 1, b = -24, and c = 16. 16. Calculate the values of x using the quadratic formula: x = (24 ± √(24^2 - 4(1)(16)))/(2(1)). 17. Simplify the equation: x = (24 ± √(576 - 64))/2. 18. Simplify further: x = (24 ± √512)/2. 19. Calculate the two possible values of x: x = (24 ± 16√2)/2. 20. Simplify: x = 12 ± 8√2.

Therefore, the two possible values for x are 12 + 8√2 and 12 - 8√2.

To find the corresponding values of y, we substitute these values of x into the equation y = (4x)/(x - 4).

Let's calculate the values of x and y:

For x = 12 + 8√2: y = (4(12 + 8√2))/((12 + 8√2) - 4) y = (48 + 32√2)/(8√2) y = 6 + 4√2

For x = 12 - 8√2: y = (4(12 - 8√2))/((12 - 8√2) - 4) y = (48 - 32√2)/(4√2) y = 12 - 8√2

Therefore, the two possible solutions are: 1. The first pipe fills the pool in approximately 12 + 8√2 hours, and the second pipe fills the pool in approximately 6 + 4√2 hours. 2. The first pipe fills the pool in approximately 12 - 8√2 hours, and the second pipe fills the pool in approximately 12 - 8√2 hours.

Please note that these values are approximate and may be rounded to a certain number of decimal places depending on the desired level of precision.

I hope this helps! Let me know if you have any further questions.