One tap fills a pool. The other one empties it. It's a word problem.In a...

dourtuntellorvl

dourtuntellorvl

Answered

2022-06-24

One tap fills a pool. The other one empties it. It's a word problem.
In a pool there are two taps, one for filling and one for emptying. Once, when the pool was empty they opened the filling tap for 4 hours. Afterwards, they opened by mistake the emptying tap and after 2 hours the pool was filled 80% from its volume. When the pool was completely filled it turned out that the filling tap filled like water amount of 1 1 2 pools and the emptying tap emptied water amount of 1 2 pool.
So what I did is
power / rate time (hours) Total Filling tap until 80% 1 x 6 1 x × 6 = 6 x Emptying tap until 80% 1 y 2 1 y × 2 = 2 y Filling tap until the end 1 x 1.5 x 1.5 Emptying tap until the end 1 y 0.5 y 0.5
I can write that:
6 x 2 y = 0.8
And here I'm stuck. I don't know how to use the information of the last two rows of my chart. If I would know the time it took to fill 100% the pool then I could sum up the two last rows from the middle column. Or the time it took to fill up the remaining 20%.
Sorry if my translation of the question wasn't so good.

Answer & Explanation

Rebekah Zimmerman

Rebekah Zimmerman

Expert

2022-06-25Added 32 answers

Let's say the time between when the pool was 80% filled and when the pool was completely filled is z hours. The filling tap had been running for 6 + z hours and the emptying tap had been running for 2 + z hours. Thus, we have the two following equations:
6 + z x = 1.5 z = 1.5 x 6
2 + z y = 0.5 z = 0.5 y 2
Set both equations equal to each other using the Transitive Property of Equality:
1.5 x 6 = 0.5 y 2 1.5 x = 0.5 y + 4 3 x = y + 8 x = y + 8 3
Substitute this into your first equation:
6 y + 8 3 2 y = 0.8 18 y + 8 2 y = 0.8
Multiply everything by 5 y ( y + 8 )
90 y ( y + 8 ) y + 8 10 y ( y + 8 ) y = 4 y ( y + 8 ) 90 y 10 ( y + 8 ) = 4 y 2 + 32 y 80 y 80 = 4 y 2 + 32 y
Put everything on one side of the equation and then divide by 4:
4 y 2 48 y + 80 = 0 y 2 12 y + 20 = 0
This quadratic can be factored into ( y 10 ) ( y 2 )
( y 10 ) ( y 2 ) = 0 y 10 = 0   OR   y 2 = 0 y = 10   OR   y = 2
Thus, y = 10 or y = 2. However, if y = 2, then, when we plug back into our z = 0.5 y 2 formula, we get z = 0, which doesn't make any sense since z is the amount of time between the pool being 80% filled and the pool being completely filled. Therefore, y = 10
Finally, substitute this into x = y + 8 3
x = 10 + 8 3 = 18 3 = 6
Therefore, the filling tap takes 6 hours to fill a pool while the emptying tap takes 10 hours to fill a pool.
(Also, if you're curious, you can plug y = 10 back into the z = 0.5 y 2 formula or plug x = 6 back into the x = 1.5 x 6 formula to find that there were 3 hours between the pool being 80% filled and the pool being completely filled, meaning this whole process took 4 + 2 + 3 = 9 hours overall.)

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