Three automated pumps, A, B, and C, can independently fill a water reservoir in 12 hours, 8 hours, and 24 hours, respectively. All three pumps start operating simultaneously. After 2 hours, pumps A and B are turned off, and pump C fills the rest of the reservoir alone. What is the total time taken to fill the entire reservoir?

The correct option is 14 hours.

To find the total time taken to fill the entire reservoir, we calculate the individual filling rates of each pump per hour, determine the work done when all pumps operate together, and then find the remaining time required for pump C to complete the filling alone.

Let the total capacity of the reservoir be equal to 1 unit.
The individual filling rates per hour for pumps A, B, and C are:
Rate of pump A = $\frac{1}{12}$ of the reservoir per hour
Rate of pump B = $\frac{1}{8}$ of the reservoir per hour
Rate of pump C = $\frac{1}{24}$ of the reservoir per hour

Step 1: Calculate the combined work done in the first 2 hours
During the first 2 hours, pumps A, B, and C operate simultaneously. Their combined filling rate per hour is:
$\text{Combined Rate}=\frac{1}{12}+\frac{1}{8}+\frac{1}{24}$

Taking the least common multiple (LCM) of 12, 8, and 24, which is 24:
$\text{Combined Rate}=\frac{2+3+1}{24}=\frac{6}{24}=\frac{1}{4}$ of the reservoir per hour

The portion of the reservoir filled in the 2 hours by all three pumps together is:
$\text{Work done in 2 hours}=2\times \frac{1}{4}=\frac{1}{2}$ of the reservoir

Step 2: Calculate the remaining portion of the reservoir
After 2 hours, the remaining portion of the reservoir to be filled is:
$\text{Remaining Work}=1-\frac{1}{2}=\frac{1}{2}$

Step 3: Calculate the time taken by pump C to fill the remaining portion
Pumps A and B are turned off after 2 hours, so pump C works alone at its rate of $\frac{1}{24}$ per hour to fill the remaining $\frac{1}{2}$ of the reservoir:
$\text{Time taken by C}=\frac{\text{Remaining Work}}{\text{Rate of C}}=\frac{1/2}{1/24}=\frac{24}{2}=12\text{hours}$

Step 4: Calculate the total time taken
The total time to fill the reservoir is the initial 2 hours when all pumps operated plus the 12 hours pump C worked alone:
$\text{Total Time}=2\text{hours}+12\text{hours}=14\text{hours}$

Hence, the total time taken to fill the entire reservoir is 14 hours.