Final answer:
It will take 4 days to fill the empty pool with both pipes open and the pump on.
Step-by-step explanation:
To find the time it will take to fill the empty pool with both pipes open and the pump on, we need to calculate the rates at which each inlet fills the pool and the rate at which the pump empties the pool.
The first inlet fills the pool in 3 days, which means it fills 1/3 of the pool in 1 day. The second inlet fills the pool in 6 days, which means it fills 1/6 of the pool in 1 day. Therefore, the combined rate at which both inlets fill the pool is 1/3 + 1/6 = 1/2 of the pool per day.
The pump empties the full pool in 4 days, which means it empties 1/4 of the pool in 1 day. Since we want to find the time it takes to fill the empty pool, we need to subtract the rate at which the pump empties the pool from the combined rate of filling the pool.
Therefore, the net rate at which the pool is filling with both inlets open and the pump on is 1/2 - 1/4 = 1/4 of the pool per day. This means that it will take 4 days to fill the empty pool with both pipes open and the pump on.