Question

Два насоса заполняют бассейн за 10 часов , причём второй насос начинает работать на 4 часа позже первого. Если бы бассейн заполнялся каждым насосом в отдельности, то первому насосу потребовало бы на 3 часа меньше, чем второму. За сколько времени может заполнить бассейн второй насос работая отдельно? Помогите решить пожалуйста)))

Answer 1

Problem Analysis

We are given that two pumps fill a pool in 10 hours, with the second pump starting 4 hours after the first pump. If each pump filled the pool separately, the first pump would take 3 hours less than the second pump. We need to find the time it takes for the second pump to fill the pool on its own.

Solution

Let's assume that the first pump fills the pool in x hours. According to the given information, the second pump would take x + 3 hours to fill the pool if it worked alone.

To find the time it takes for the first pump to fill the pool, we can set up the following equation based on the given information:

1/x + 1/(x+3) = 1/10

Simplifying this equation will give us the value of x, which represents the time it takes for the first pump to fill the pool.

Calculation

To solve the equation, we can multiply both sides by 10x(x+3) to eliminate the denominators:

10(x+3) + 10x = x(x+3)

Expanding and simplifying the equation:

10x + 30 + 10x = x^2 + 3x

20x + 30 = x^2 + 3x

Rearranging the equation:

x^2 + 3x - 20x - 30 = 0

x^2 - 17x - 30 = 0

Now we can solve this quadratic equation to find the value of x.

Using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = -17, and c = -30.

Calculating the discriminant:

D = b^2 - 4ac = (-17)^2 - 4(1)(-30) = 289 + 120 = 409

Since the discriminant is positive, we have two real solutions for x.

Using the quadratic formula:

x = (-(-17) ± √(409)) / 2(1)

Simplifying:

x = (17 ± √409) / 2

Calculating the two possible values of x:

x1 = (17 + √409) / 2 ≈ 16.61 x2 = (17 - √409) / 2 ≈ 0.39

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

Therefore, the time it takes for the first pump to fill the pool is approximately 16.61 hours.

To find the time it takes for the second pump to fill the pool on its own, we subtract 4 hours from the time it takes for the first pump:

16.61 - 4 ≈ 12.61

Therefore, the second pump can fill the pool on its own in approximately 12.61 hours.

Answer

The second pump can fill the pool on its own in approximately 12.61 hours.