Final answer:
It will take 3 hours to fill the fountain with both sprinklers operating.
Step-by-step explanation:
To find out how long it will take to fill the fountain with both sprinklers operating, we need to calculate the combined rate at which the two sprinklers can fill the fountain. Let's call the rate at which the first sprinkler can fill the fountain R1 and the rate at which the second sprinkler can fill the fountain R2.
The first sprinkler can fill the fountain in 4 hours, so its rate is 1/4 of the fountain per hour (1 fountain/4 hours = 1/4 fountain/hour = R1).
The second sprinkler can fill the fountain in 12 hours, so its rate is 1/12 of the fountain per hour (1 fountain/12 hours = 1/12 fountain/hour = R2).
The combined rate at which the two sprinklers can fill the fountain is R1 + R2. In this case, it is (1/4 + 1/12) fountain/hour = (3/12 + 1/12) fountain/hour = 4/12 fountain/hour = 1/3 fountain/hour. Therefore, it will take the two sprinklers operating together 3 hours to fill the fountain.