I need help with a word problem: One pipe can fill a hot tub in 9 minutes, while a second pipe can fill it in 15 minutes. If the tub is empty, how long will it take both pipes together to fill the hot tub?
I have several like this, so can you show me the formula so I can figure out how to plug the others in?
Jul 28th 2005, 01:23 AM
Let's see... By the data of the problem, I suppose the pipes supply water at a constant rate. For the first pipe, call this rate . If the pipe supplies units of water in time t minutes, then we have , which integrates to the term being the initial supply at time . I guess , as we have to set the timer and open the tap simultaneously ( :confused: :rolleyes: :eek: :p ). If the tub fills at units of water, then , and so . We conclude that, for the first pipe, the supply is .
Under the same suicidal considerations, we obtain that the supply of the second pipe is . For both pipes, the supply is (almost). When the tub fills, we will have , from where minutes...
The tub is full. Now, lets give president Bush a bath. :mad:
Jul 28th 2005, 01:56 AM
Here is one way.
So you have several questions like that. That is because that is a popular question.
And, I think there is a popular formula already developed for that type of question. I think it is
1/x +1/y = 1/t --------******
x = time the task can be done/finished alone by one performer.
y = another time the task can be done/finished alone also by another performer.
t = the time the task can be done/finished if the two performers perform together.
We can derive that formula, why not.
Let J = complete task to be done.
and, performer A can finish it alone in x time.
and, performer B can finish it also alone in y time.
and, if A and B perform together, the J will be finished in t time.
Like distance, task = rate*time
So, rate = task/time ------***
Rate of A alone is J/x.
Rate of B alone is J/y.
If A and B perform together, their combined rate is (J/x +J/y).
task = rate*time
J = (J/x +J/y)*t
Divide both sides by t,
J/t = J/x +J/y
Divide both sides by J,
1/t = 1/x +1/y
1/x +1/y = 1/t -------the formula.
Now, per your example,
x = 9 min.
y = 15 min
1/x +1/y = 1/t
1/9 +1/15 = 1/t
Clear the fractions, multiply both sides by 9*15*t,
1*15*t +1*9*t = 1*9*15
15t +9t = 135
24t = 135
t = 135/24
t = 5.625 min ---- the time the tub will fill up if the two pipes will do it together.
Wait, there is a shorter formula, or a simplification of the derived formula above;
1/x +1/y = 1/t
ty +tx = x*y
t = (x*y) / (x+y) -----shorter formula.
t = (9*15)/(9+15) = 135/24 = 5.625 min----same.
One big problem if you rely only on formulas, once you forget the formula then you are lost.