<p>How will long it take to empty the tank if both the inlet pipe P1 and the outlet pipe P2 are opened simultaneously?</p> <p>(i) 𝑃<sub>1</sub> can fill the tank in 16 minutes.</p> <p>(ii) 𝑃<sub>2</sub> can empty the full tank in 8 minutes.</p>
This problem can be solved by calculating the rate at which each pipe works and then finding their combined rate. The rate is the fraction of the tank that is filled or emptied per minute.
When both pipes are open, their rates are added together to find the net rate of change for the water in the tank.
Combined Rate = RateP1 + RateP2
Combined Rate = (1/16) + (-1/8)
To add these fractions, we find a common denominator (16):
Combined Rate = 1/16 - 2/16 = -1/16 of the tank per minute.
The combined rate is -1/16. The negative sign confirms that the tank is emptying. The magnitude, 1/16, means that 1/16th of the tank is emptied every minute.
To find the total time to empty the full tank, we take the reciprocal of this rate:
Time = 1 / (1/16) = 16 minutes.
It will take 16 minutes to empty the tank if both pipes are opened simultaneously.
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