Two pumps are filling a pool. One of them is high power and can fill the pool alone in 2 hours less time than the other can do so. Given that, working together, both pumps can fill the pool in 144 minutes, how long, in hours, will it take the powerful pump to fill the pool alone?
The answer is 4 hours
s = smaller pump fill time ; b = bigger pump fill time
s = b + 120
(144 / s) + (144 / b) = 1
[144 / (b + 120)] + (144 / b) = 1
144 b + 144 b + (144 * 120) = b (b + 120)
0 = b^2 - 168 b - 17280
convert answer to hours
To solve this problem, let's consider the rates at which each pump fills the pool.
Let's assume that the slower pump takes "x" hours to fill the pool alone. Therefore, the faster pump will take "x - 2" hours to fill the pool alone based on the given information.
Now, let's consider the rates at which each pump fills the pool. The slower pump fills at a rate of 1/x of the pool per hour, and the faster pump fills at a rate of 1/(x-2) of the pool per hour.
If both pumps work together, their combined rate is the sum of their individual rates. So their combined rate is (1/x) + (1/(x-2)) of the pool per hour.
We know that together, they can fill the pool in 144 minutes, which is equal to 144/60 = 2.4 hours.
So, we can set up the equation:
(1/x) + (1/(x-2)) = 1/2.4
To solve this equation, we can multiply both sides by the lowest common denominator, which is 2.4x(x-2), to get rid of the fractions:
2.4(x-2) + 2.4x = x(x-2)
Expand and simplify the equation:
2.4x - 4.8 + 2.4x = x^2 - 2x
Combine like terms:
4.8x - 4.8 = x^2 - 2x
Rearrange the equation:
x^2 - 6.8x + 4.8 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, factoring may not yield nice integer solutions, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = -6.8, and c = 4.8. Substituting these values into the quadratic formula and calculating, we find:
x = (6.8 ± √((-6.8)^2 - 4*1*4.8)) / (2*1)
x = (6.8 ± √(46.24 - 19.2)) / 2
x = (6.8 ± √27.04) / 2
x ≈ 5.9 or x ≈ 0.9
Since we are interested in the positive value for x, which represents the time taken by the slower pump, we have x ≈ 5.9.
Therefore, the powerful pump will take approximately 5.9 - 2 = 3.9 hours (or 3 hours and 54 minutes) to fill the pool alone.
1/x + 1/(x-2) = 60/144
now just solve for x