Alright Eli, let's work through this step by step and complete the table you have.
First, let's define what we have and need to find using the variables given:
1. 10% Acetic Acid Solution
- Volume: [tex]\(0.5\)[/tex] gallons
- Concentration: [tex]\(10\%\)[/tex] (or 0.10)
2. 35% Acetic Acid Solution
- Volume: [tex]\(v\)[/tex] gallons
- Concentration: [tex]\(35\%\)[/tex] (or 0.35)
3. Mixture
- Desired Concentration: [tex]\(15\%\)[/tex] (or 0.15)
- Total Volume: [tex]\(v + 0.5\)[/tex] gallons
Now, let's calculate the amounts as requested and complete the table. Here's the additional information needed for calculations:
- The amount of acetic acid in the [tex]\(10\%\)[/tex] solution is:
[tex]\[
0.5 \text{ gallons} \times 0.10 = 0.05 \text{ gallons of acetic acid}
\][/tex]
- The amount of acetic acid in the [tex]\(35\%\)[/tex] solution is:
[tex]\[
0.35 v \text{ gallons of acetic acid}
\][/tex]
- The total volume of the mixture:
[tex]\[
v + 0.5 \text{ gallons}
\][/tex]
- The amount of acetic acid in the mixture:
[tex]\[
0.15 (v + 0.5) \text{ gallons of acetic acid}
\][/tex]
Given that we want the total amount of acetic acid in the mixture to be the sum of the amount from the [tex]\(10\%\)[/tex] and [tex]\(35\%\)[/tex] solutions, we can set up the equation:
[tex]\[
0.05 + 0.35 v = 0.15 (v + 0.5)
\][/tex]
From the values calculated:
[tex]\[
v = 0.125 \text{ gallons}
\][/tex]
Now, let's fill out the required table with these values:
\begin{tabular}{|c|c|c|c|}
\hline \begin{tabular}{c}
Amount of \\
Solution
\end{tabular} & \begin{tabular}{c}
Acid \\
Concentration
\end{tabular} & \begin{tabular}{c}
Amount of \\
Acid
\end{tabular} \\
\hline
[tex]$10\%$[/tex] Acid & [tex]$0.5$[/tex] gal & [tex]$0.10$[/tex] & [tex]$(0.10)(0.5) = 0.05$[/tex] \\
\hline
[tex]$35\%$[/tex] Acid & [tex]$v = 0.125$[/tex] gal & [tex]$0.35$[/tex] & [tex]$0.35 (0.125) = 0.04375$[/tex] \\
\hline
Mixture & [tex]$v + 0.5 = 0.625$[/tex] gal & [tex]$0.15$[/tex] & [tex]$0.15 (0.625) = 0.09375$[/tex] \\
\hline
\end{tabular}
So, the filled table is:
> \begin{tabular}{|c|c|c|c|}
> \hline & \begin{tabular}{c}
> Amount of \\
> Solution
> \end{tabular} & \begin{tabular}{c}
> Acid \\
> Concentration
> \end{tabular} & \begin{tabular}{c}
> Amount of \\
> Acid
> \end{tabular} \\
> \hline [tex]$10\%$[/tex] Acid & [tex]$0.5$[/tex] gal & [tex]$0.10$[/tex] & [tex]$0.05$[/tex] \\
> \hline [tex]$35\%$[/tex] Acid & [tex]$0.125$[/tex] gal & [tex]$0.35$[/tex] & [tex]$0.04375$[/tex] \\
> \hline Mixture & [tex]$0.625$[/tex] gal & [tex]$0.15$[/tex] & [tex]$0.09375$[/tex] \\
> \hline
> \end{tabular}
This table now accurately represents the given problem and the quantities involved.
