the fill pipe for a tank can fill the tank in 4 hours and the drain pipe can drain in 2 hours. If both pipes are accidentally opened, how long will it take to empty a half-filled tank?
Let V = tank volume
Fill rate = 1/4 V per hr
Drain rate = 1/2 V per hr
Net loss rate = 1/2 - 1/4 = 1/4 V/hr
Drain time = (V/2)/(1/4 V/hr) = 2 hr
To determine how long it will take to empty a half-filled tank when both the fill pipe and drain pipe are open, we need to calculate the net rate at which the tank is being drained.
Let's first find the rates of each pipe:
- Fill Pipe: It can fill the entire tank in 4 hours, so its rate is 1/4 tank per hour.
- Drain Pipe: It can drain the entire tank in 2 hours, so its rate is 1/2 tank per hour.
Now, let's calculate the net rate of draining when both pipes are open:
Since the fill pipe is trying to fill the tank and the drain pipe is trying to empty it simultaneously, we subtract the rate of the drain pipe (1/2 tank per hour) from the rate of the fill pipe (1/4 tank per hour):
1/4 tank per hour - 1/2 tank per hour = -1/4 tank per hour.
The negative sign indicates that the tank is being emptied.
Now, since the net rate of draining is -1/4 tank per hour, it means that the tank will be emptied by -1/4 tank every hour.
To determine how long it will take to empty a half-filled tank, which is 1/2 tank, we can use the formula:
Time = Amount / Rate.
In this case, the amount is 1/2 tank, and the rate is -1/4 tank per hour. Therefore:
Time = (1/2) / (-1/4)
Time = (1/2) * (-4/1)
Time = -2 hours.
The negative sign indicates that the tank is being emptied, so the answer is 2 hours.
Hence, it will take 2 hours to empty a half-filled tank when both the fill pipe and the drain pipe are accidentally opened.