A tank has a supply pipe P and an exhaust pipe Q. The pipe P takes 4 minutes less time to fill the tank than the pipe Q to empty it. When both are open, the tank is filled in 24 minutes. Find the time needed for the pipe P to fill the tank when Q is closed.

Question

phi · Answer

I would say the tank is emptied at a rate of $$ \frac{1 \text{ tank}}{M \text{ mins}} $$
and is filled at a rate of
$$ \frac{1 \text{ tank}}{(M-4) \text{ mins}} $$

Now use rate * time = # of tanks filled  (1 in this case)
if we both fill and empty the tank at the same time the combined rate is
$$ \frac{1}{M-4} - \frac{1}{M} $$
if we do this for 24 minutes, we get a filled tank:
$$ 24\left(\frac{1}{M-4} - \frac{1}{M}\right) = 1 $$

can you solve for M ?

anonymous · Answer

@phi M = -8, 12

Since there is no negative time, 12 minutes is the answer...?

phi · Answer

We have to be careful.  I picked M to be the time needed to empty the tank (Probably should have defined M to be the time to fill the tank).

so the tank empties in 12 minutes.  

The tank fills at a rate of 
$$ \frac{1}{M-4} = \frac{1 \text{ tank}}{8 \text{ minutes} } $$

that means it takes 8 minutes to fill the tank.
or, using algebra
$$  \frac{1 \text{ tank}}{8 \text{ minutes} }\cdot t = 1 \text{ tank} \
t = 1 \cancel{\text{ tank}}\cdot \frac{8 \text{ minutes} }{1  \cancel{\text{ tank}}}\
t= 8 \text{ minutes}
$$

anonymous · Answer

@phi  Thank you!

phi · Answer

yw