When two hoses are connected to a swimming pool, they can fill the pool in 4 hours if both are open. If only the larger hose is open, it can fill the pool in 6 hours less than the smaller hose. How long will it take tge smaller hose to fill the swimming pool if it works alone?
Use quadratic equation
Someone solve it pls i need help :(
1/x + 1/(x-6) = 1/4
x = 12
Give exact solution
To solve this problem using the quadratic equation, let's assign variables to the unknowns. Let x represent the time (in hours) it would take for the smaller hose to fill the pool alone.
According to the given information:
- When both hoses are open, they can fill the pool in 4 hours.
- If only the larger hose is open, it can fill the pool in 6 hours less than the smaller hose.
We can set up the following equations:
1. When both hoses are open:
1/x + 1/(x-6) = 1/4
2. If only the larger hose is open:
1/(x-6) = 1
Now, to solve the quadratic equation:
1. Multiply both sides of equation 1 by 4*x*(x-6) to eliminate the denominators:
4*(x-6) + 4*x = x*(x-6)
2. Simplify equation 1:
4x - 24 + 4x = x^2 - 6x
3. Combine like terms:
8x - 24 = x^2 - 6x
4. Rearrange equation 3 to form a quadratic equation:
x^2 - 14x + 24 = 0
Now, we can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 1, b = -14, and c = 24:
x = (-(-14) ± √((-14)^2 - 4(1)(24))) / (2(1))
Simplifying further:
x = (14 ± √(196 - 96)) / 2
x = (14 ± √100) / 2
x = (14 ± 10) / 2
Now, we can find the two possible values for x:
x1 = (14 + 10) / 2 = 12
x2 = (14 - 10) / 2 = 2
Since we're solving for time, we ignore the negative value x2 = 2, as time cannot be negative.
Therefore, the time it would take for the smaller hose to fill the swimming pool alone is 12 hours.