Solving Using Rates

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.

1. Determine the Rate of Each Pipe

• Inlet Pipe P1: Fills the tank in 16 minutes. Its rate is positive (filling).
  Rate_P1 = +1/16 of the tank per minute.
• Outlet Pipe P2: Empties the tank in 8 minutes. Its rate is negative (emptying).
  Rate_P2 = -1/8 of the tank per minute.

2. Calculate the Combined Rate

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 = Rate_P1 + Rate_P2

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.

3. Calculate the Time to Empty the Tank

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.

Conclusion

It will take 16 minutes to empty the tank if both pipes are opened simultaneously.