a tank has 2 inlets P and Q and an outlet tap R .
when empty the tank can be filled by tap P alone in 4 and a half hours
So tap P's filling rate is 1 tank per 4 1/2 hours or
1 tank per 9/2 hours or 1/(9/2) tank per hour or 2/9 tank per hour
or by tap Q alone in 3 hours.
So tap P's filling rate is 1 tank per 3 hours or 1/3 tank per hour
when full the tank can be emptied in 2 hours by tap R
So tap R's emptying rate (negative filling rate) is -1 tank per 2 hours or
-1/2 tank per hour
a) the tank is initially empty . find how long it would take to fill up the
tank i)if tap R is closed and taps P and Q are opened at the same time.
Their combined rate is 2/9 + 1/3 = 2/9 + 3/9 = 5/9 tank/hour
In x hours they will fill 1 tank:
(5/9)x = 1
x = 9/5 = 1.8 hours = 1 hour and 48 minutes.
ii) if all the three taps are opened at the same time.
Their combined rate is 2/9 + 1/3 - 1/2 = 4/18 + 6/18 - 9/18 = 1/9 tank/hour
In x hours they will fill 1 tank:
(1/9)x = 1
x = 9 hours.
B) the tank is initially empty and the three taps are opened as follows:
P at 8:00 A.m Q at 8.45 A.m R at 9.00 A.m
i)find the fraction of the tank that would be filled by 9.00 a.m
P has been open 1 hour at 9AM and since it fills 2/9 tank per hour it
has filled 2/9 of the tank.
Q has been open 15 minutes or 1/4 hour at 9AM and since it fills 1/3 of
a tank per hour it has filled 1/4 of 1/3 or 1/12 tank.
R hasn't opend yet, so the fraction of a tank that has been filled is
2/9 + 1/12 = 8/36 + 3/36 = 11/36 of a tank.
ii)find the time the tank would be fully filled up.
It is 11/36 full at 9AM so it has 25/36 tank more to go, so in x hours after
9AM it will fill the remaining 25/36 tank at the combined rate of 1/9 tank
per hour, so
(1/9)x = 25/36
x = (25/36)(9/1) = 25/4 = 6 1/4 hours
6 hours 15 minutes after 9AM is 6:15PM.
Edwin
