A swimming pool is filled with 8,520 gallons of water. The pool begins to drain at a rate of 12 gallons per minute. Which equation can be used to find the number of minutes, $m$, it will take for the pool to be half full?

A$8,520-12 m=4,260$
B$-8,520+12 m=4,260$
C$8,520-12 m=\frac{1}{2} m$
D$8,520-\frac{1}{2}(12 m)=0$

The swimming pool contains an initial volume of $8{,}520$ gallons of water. It drains at a constant rate of $12$ gallons per minute. Let $m$ represent the time in minutes that has elapsed since the pool began draining.

After $m$ minutes, the total amount of water that has drained from the pool is $12 \times m$, or $12m$. Therefore, the amount of water remaining in the pool after $m$ minutes is given by the expression:
\[8{,}520 - 12m\]

Setting Up the Equation

The objective is to determine when the pool is half full. The total initial volume is $8{,}520$ gallons, so half of that volume is:
\[\frac{8{,}520}{2} = 4{,}260 \text{ gallons}.\]
To find the time $m$ at which the pool contains exactly half of its initial capacity, we set the expression for the remaining water equal to this target volume:
\[8{,}520 - 12m = 4{,}260\]

Evaluating the Options and Conclusion

We examine the given options to find the one that matches our derived equation:

Option A: $8{,}520 - 12m = 4{,}260$
Option B: $-8{,}520 + 12m = 4{,}260$
Option C: $8{,}520 - 12m = \frac{1}{2}m$
Option D: $8{,}520 - \frac{1}{2}(12m) = 0$

Our derived equation $8{,}520 - 12m = 4{,}260$ is identical to Option A. Therefore, this is the correct equation to find the time $m$.

Examples

This type of algebraic modeling is essential for managing resources. For example, if you are monitoring the fuel in a tank or tracking your budget over time, you can model the remaining amount as an initial quantity minus the rate of expenditure times time. Using equations to find when a target level is reached helps in planning maintenance, predicting shortages, or budgeting effectively.