Two hoses are connected to a swimming pool. Working together, they can fill the pool in 4 hr. The larger hose, working alone, can fill the pool in 6 hr less time than the smaller one. How long would it take the smaller one, woking alone, to fill the pool?
Please explain
If small = s, big = b, sb = both, then
1/sb = 1/s + 1/b
1/4 = 1/s + 1/6
1/12 = 1/s
s = 12 hours
Thank you!
Yeah. Too bad I did it wrong. I read "6 hours" rather than "6 hours less".
It's been done correctly later on.
See the solution to the problem posted Nov 5 at 10:18 pm
To solve this problem, let's start by assigning some variables. Let "x" represent the amount of time it takes for the smaller hose to fill the pool on its own.
We are given that the two hoses can fill the pool together in 4 hours. This means that the combined rate of the two hoses is 1 pool per 4 hours, or 1/4 pool per hour.
We are also given that the larger hose alone can fill the pool in 6 hours less time than the smaller hose. So, the larger hose takes x - 6 hours to fill the pool.
Now, let's use the concept of rates to set up an equation. The rate at which each hose fills the pool can be found by dividing the amount of work (in this case, filling the pool) by the time taken.
The rate of the smaller hose alone is 1 pool / x hours.
The rate of the larger hose alone is 1 pool / (x - 6) hours.
When the two hoses work together, their rates add up, so the equation becomes:
1/x + 1/(x-6) = 1/4
To solve this equation, we can multiply both sides by the least common multiple (LCM) of the denominators, which in this case is 4x(x-6). This gives us:
4(x-6) + 4x = x(x-6)
Now, we can simplify and solve for x:
4x - 24 + 4x = x^2 - 6x
8x - 24 = x^2 - 6x
0 = x^2 - 14x + 24
To factor this quadratic equation, we can find two numbers that multiply to give 24 and add up to -14. These numbers are -2 and -12, so the equation factors as:
(x - 2)(x - 12) = 0
Setting each factor equal to zero, we find two possible values for x:
x - 2 = 0, which gives x = 2
x - 12 = 0, which gives x = 12
Since we are looking for the time it takes for the smaller hose to fill the pool alone, we discard the solution x = 2 because it would mean that the larger hose fills the pool in -4 hours.
Therefore, the answer is x = 12. It would take the smaller hose, working alone, 12 hours to fill the pool.