Sure, let's carefully analyze and derive the equation that Grant can use to find the number of gallons of water, [tex]\( n \)[/tex], that he should remove.
### Step-by-Step Solution:
1. Initial Setup:
- Initial volume of the solution: 22 gallons.
- Initial concentration of ammonia: 16% (or 0.16).
- Desired concentration of ammonia: 24% (or 0.24).
2. Calculate the initial amount of ammonia in the solution:
The initial amount of ammonia can be found using the initial concentration and volume:
[tex]\[
\text{Initial amount of ammonia} = \text{initial concentration} \times \text{initial volume}
\][/tex]
[tex]\[
\text{Initial amount of ammonia} = 0.16 \times 22
\][/tex]
Simplifying this:
[tex]\[
0.16 \times 22 = 3.52 \quad \text{(gallons of ammonia)}
\][/tex]
3. Remaining amount of solution after removing [tex]\( n \)[/tex] gallons of water:
If [tex]\( n \)[/tex] gallons of water are removed from the solution, the remaining volume of the solution will be:
[tex]\[
22 - n \quad \text{(gallons)}
\][/tex]
4. Create an equation for the new concentration:
The final concentration of ammonia in the solution is 24% (or 0.24). The volume of the remaining solution is [tex]\( 22 - n \)[/tex] gallons.
Using the definition of concentration:
[tex]\[
\text{Concentration} = \frac{\text{Amount of solute}}{\text{Volume of solution}}
\][/tex]
We set up the equation representing the final state of the solution:
[tex]\[
0.24 = \frac{\text{Amount of ammonia}}{22 - n}
\][/tex]
5. Substitute the known amount of ammonia:
From step 2, we know the amount of ammonia initially is 3.52 gallons. Since the ammonia does not evaporate, this remains the same in the final solution:
[tex]\[
0.24 = \frac{3.52}{22 - n}
\][/tex]
6. Rearrange to form a solvable equation:
To isolate [tex]\( n \)[/tex], multiply both sides by [tex]\( 22 - n \)[/tex]:
[tex]\[
0.24 \times (22 - n) = 3.52
\][/tex]
Hence, the equation that Grant can use to find the number of gallons of water he should remove is:
[tex]\[
3.52 = 0.24 \times (22 - n)
\][/tex]
This matches the third option provided:
[tex]\[
3.52 = 0.24 \times (22 - n)
\][/tex]
So, the correct choice is:
[tex]\[
3.52 = 0.24 \times (22 - n)
\][/tex]
