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Question 884577: It takes 30 minutes to empty a half-full tank by draining it at a constant rate. It is decided to simultaneously pump water into the half-full tank while draining it. What is the rate at which water has to be pumped in so that it gets fully filled in 10 minutes?
(A) 4 times the draining rate
(B) 3 times the draining rate
(C) 2.5 times the draining rate
(D) 2 times the draining rate
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i believe the answer will be 4 times the draining rate.
the equation to use is RT = Q
Q is equal to 1/2 the tank.
T is equal to 30 minutes
R is derived from the equation as follows:
RT = Q
R*30 = 1/2
R = 1/60
the tank is being drained at the rate of 1/60 of the tank per minute.
at that rate, the remaining 1/2 of the tank will be drained in 30 minutes.
the rate of filling and the rate of draining are opposing forces so you need to subtract one from the other to get the rate of filling.
you want to fill 1/2 the tank while at the same time you are draining the tank.
the combined formula would be as follows:
(RF - RD) * 10 = 1/2
you know RD is 1/60 because we just solved for that.
your formula becomes:
(RF - 1/60) * 10 = 1/2
simplify this to get:
10*RF - 10/60 = 1/2
add 10/60 to both sides of this equation to get:
10*RF = 1/2 + 10/60 which becomes:
10*RF = 30/60 + 10/60 which becomes:
10*RF = 40/60.
divide both sides of this equation by 10 to get:
RF = 4/60.
4/60 is 4 times 1/60.
fill rate is 4 times the drain rate.
you start at 1/2 a tank which is the same as 3/6 of a tank.
in 10 minutes you will have drained 10*1/60 = 10/60 = 1/6 more of the tank.
in the same 10 minutes you will have filled 10*4/60 = 40/60 = 4/6 of the tank.
sum of fill and drain is equal to 3/6 - 1/6 + 4/6 which is equal to 6/6 which is equal to 1.
at the end of the 10 minutes, the tank is full.
you could have solved this another way as well.
you still had to find the drain rate which is equal to 1/60 of the tank per minute.
in 10 minutes, you will have drained 1/6 more of the tank.
the tank is now 1/2 - 1/6 = 3/6 - 1/6 = 2/6 full.
in order to fill the tank in the same 10 minutes, you have to fill 4/6 of the tank.
the same formula is used.
RT=Q
10R = 4/6
R = 4/60.
your drain rate was 1/60
your fill rate is 4/60.
the fill rate is 4 times the drain rate.
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