Paul has 3 large identical empty open-top containers (C1, C2, C3) outside.

- He walks outside and soon after, at time 0 hours, a steady rain begins. Rain enters the containers at R gallons per minute (gpm).
- 7 minutes later, Paul starts emptying out C1 at a rate of J gpm. Exactly when C1 is empty, Paul starts emptying C2 out and finishes 42 minutes later.
- Exactly when he's done emptying out C2, it starts raining twice as hard (2R gpm) and Paul starts emptying out C3 twice as fast (at 2J gpm).
-When Paul finishes emptying C3, the rain stops.

1) At what time did Paul start emptying C3?
2) At what time did it stop raining?

It is given that C1 was filled 7 minutes at a rate of R gallons/minute; thus, C1 had been filled with 7R gallons of rainwater.

It is given and deduced that C2 was filled at a rate of R gallons/minute, and 7 minutes plus the time it took to empty C1 of the 7R gallons at a rate of J gallons/minute.  Thus, for 7(R/J + 1) minutes at R gallons/minute, C2 had been filled with 7R(R/J + 1) gallons of rainwater.

It is given that it took 42 minutes to empty C2. Assuming that C2 was also emptied at the rate of J gallons/minute:
(7/J)R² + 7R gallons = 42 minutes * J gallons/minute; I.e.,
(1/J)R² + R - 6J = 0.  Using the quadratic formula to find the positive root of R, it is found that R is equal to 2J.  Hence, J = R/2.  Therefore, in terms of R, C2 had been filled with (7R²/(R/2) + 7R) = 21R gallons of rainwater.  And, thus, at a rate of R gallons/minute, it took 21 minutes to fill C2.

It is given and deduced that the time it took to fill C3 was equivalent to the time it took to fill and empty C2. Thus, C3 took (21 + 42) = 63 minutes to fill. At a rate of R gallons/minute, C3 had been filled with 63R gallons of rainwater.  Therefore...
the time Paul started emptying C3 was 1 hour and 3 minutes after it started to rain.

It is given that C3 was emptied at a rate of 2J gallons/minute. In terms of R, this is R gallons/minute.  Thus, At 63R gallons, it took 63 minutes to empty C3; and the total time it took to fill and empty C3 was (63 + 63) = 126 minutes.  Therefore...
the time it stopped raining was 2 hours and 6 minutes after it started to rain.

Edited on November 6, 2008, 8:12 am

Posted by Dej Mar on 2008-11-06 01:41:17