Question

Первая труба наполняет резервуар на 45 минут дольше чем вторая,обе трубы одновременно заполняют резервуар за 14 минут,за сколько минут резервуар наполняет только вторая труба

Answer 1

Х - наполняет вторая
х+45 - первая
1 - весь резервуар
1/14 часть наполняют обе трубы за 1 мин
1/х часть наполняет вторая за 1 мин
1/(х+45) наполняет первая
1/х+1/(х+45)=1/14  (умножим на 14х(х+45))
14(х+45)+14х=х(х+45)
14х+630+14х=х^2+45х
х^2+17х-630=0
D=17*17-4(-630)=289+2520=2809  Корень из D=53
х(1)=(-17-53):2=-70:2=-35  (не подходит)
х(2)=(-17+53):2=36:2=18 (мин)
Ответ: вторая труба наполнит резервуар за 18 минут

Answer 2

Problem Analysis

We are given that the first pipe fills the reservoir 45 minutes longer than the second pipe. Both pipes together can fill the reservoir in 14 minutes. We need to determine how long it would take for the second pipe to fill the reservoir on its own.

Solution

Let's assume that the second pipe takes x minutes to fill the reservoir on its own.

From the given information, we can set up the following equations:

Equation 1: The first pipe fills the reservoir in (x + 45) minutes. Equation 2: Both pipes together fill the reservoir in 14 minutes.

To solve this system of equations, we can use the concept of rates. The rate at which the first pipe fills the reservoir is 1/(x + 45) (since it takes (x + 45) minutes to fill the reservoir). The rate at which the second pipe fills the reservoir is 1/x (since it takes x minutes to fill the reservoir).

According to the problem, when both pipes work together, their combined rate is 1/14 (since they fill the reservoir in 14 minutes).

Using the concept of rates, we can set up the following equation:

Equation 3: 1/(x + 45) + 1/x = 1/14

To solve this equation, we can multiply through by the least common multiple of the denominators (14x(x + 45)) to eliminate the fractions:

14x + 14(x + 45) = x(x + 45)

Simplifying the equation:

14x + 14x + 630 = x^2 + 45x

Rearranging the equation:

x^2 + 45x - 28x - 630 = 0

x^2 + 17x - 630 = 0

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

Quadratic Equation Solution

Using the quadratic formula, we have:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation x^2 + 17x - 630 = 0, the coefficients are: a = 1 b = 17 c = -630

Plugging these values into the quadratic formula:

x = (-17 ± √(17^2 - 4(1)(-630))) / (2(1))

Simplifying:

x = (-17 ± √(289 + 2520)) / 2

x = (-17 ± √2809) / 2

x = (-17 ± 53) / 2

We have two possible solutions for x:

x1 = (-17 + 53) / 2 = 36 / 2 = 18 x2 = (-17 - 53) / 2 = -70 / 2 = -35

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

Answer

Therefore, the second pipe will fill the reservoir on its own in 18 minutes.