To determine how long it will take to fill the pool when both the inlet and outlet pipes are left open, we first need to find the rates at which each pipe works.
-
Inlet pipe filling rate: The inlet pipe can fill the pool in 9 hours. Therefore, its rate is: \[ \text{Rate of inlet pipe} = \frac{1 \text{ pool}}{9 \text{ hours}} = \frac{1}{9} \text{ pools per hour} \]
-
Outlet pipe emptying rate: The outlet pipe can empty the pool in 12 hours. Thus, its rate is: \[ \text{Rate of outlet pipe} = \frac{1 \text{ pool}}{12 \text{ hours}} = \frac{1}{12} \text{ pools per hour} \]
-
Net rate when both pipes are open: The combined rate when the inlet and outlet pipes are both open is the rate of the inlet minus the rate of the outlet: \[ \text{Net rate} = \frac{1}{9} - \frac{1}{12} \]
To compute this, we need a common denominator. The least common multiple of 9 and 12 is 36. So, we can express both rates with a denominator of 36: \[ \frac{1}{9} = \frac{4}{36} \quad \text{and} \quad \frac{1}{12} = \frac{3}{36} \] Now, substituting these into the net rate equation: \[ \text{Net rate} = \frac{4}{36} - \frac{3}{36} = \frac{1}{36} \text{ pools per hour} \]
- Time to fill the pool: To find out how long it will take to fill one pool at this net rate, we can take the reciprocal of the net rate: \[ \text{Time} = \frac{1 \text{ pool}}{\frac{1}{36} \text{ pools per hour}} = 36 \text{ hours} \]
Thus, it will take 36 hours to fill the pool when both pipes are left open.