Let's complete the chart step-by-step and then determine the quantity at which the business firm is maximizing its profits.
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Quantity} & \text{Price} & \text{Total Revenue} & \text{Marginal Revenue} & \text{Total Cost} & \text{Marginal Cost} & \begin{tabular}{l}
\text{Profit or Loss} \\
(\text{TR - TC})
\end{tabular} \\
\hline
1 & \$20 & \$20 & - & \$14 & - & \$6 \\
\hline
2 & \$19 & \$38 & \$18 & \$24 & \$10 & \$14 \\
\hline
3 & \$18 & \$54 & \$16 & \$39 & \$15 & \$15 \\
\hline
4 & \$17 & \$68 & \$14 & \$61 & \$22 & \$7 \\
\hline
5 & \$16 & \$80 & \$12 & \$95 & \$34 & -\$15 \\
\hline
\end{array}
\][/tex]
Let's go through the derivations for each element step-by-step:
1. Total Revenue (TR):
[tex]\[
\begin{align*}
\text{TR}_1 & = 1 \times 20 = 20 \\
\text{TR}_2 & = 2 \times 19 = 38 \\
\text{TR}_3 & = 3 \times 18 = 54 \\
\text{TR}_4 & = 4 \times 17 = 68 \\
\text{TR}_5 & = 5 \times 16 = 80
\end{align*}
\][/tex]
2. Marginal Revenue (MR):
[tex]\[
\begin{align*}
\text{MR}_2 & = 38 - 20 = 18 \\
\text{MR}_3 & = 54 - 38 = 16 \\
\text{MR}_4 & = 68 - 54 = 14 \\
\text{MR}_5 & = 80 - 68 = 12
\end{align*}
\][/tex]
3. Total Cost (TC):
[tex]\[
\begin{align*}
\text{TC}_1 & = 14 \\
\text{TC}_2 & = 24 \\
\text{TC}_3 & = 39 \\
\text{TC}_4 & = 61 \\
\text{TC}_5 & = 95
\end{align*}
\][/tex]
4. Marginal Cost (MC):
[tex]\[
\begin{align*}
\text{MC}_2 & = 24 - 14 = 10 \\
\text{MC}_3 & = 39 - 24 = 15 \\
\text{MC}_4 & = 61 - 39 = 22 \\
\text{MC}_5 & = 95 - 61 = 34
\end{align*}
\][/tex]
5. Profit or Loss (TR - TC):
[tex]\[
\begin{align*}
\text{Profit or Loss}_1 & = 20 - 14 = 6 \\
\text{Profit or Loss}_2 & = 38 - 24 = 14 \\
\text{Profit or Loss}_3 & = 54 - 39 = 15 \\
\text{Profit or Loss}_4 & = 68 - 61 = 7 \\
\text{Profit or Loss}_5 & = 80 - 95 = -15
\end{align*}
\][/tex]
Now, we have all the necessary values filled in the chart.
The profit or loss values for each quantity are: \[tex]$6, \$[/tex]14, \[tex]$15, \$[/tex]7, -\$15.
Profit is maximized at the quantity where the profit value is highest. Therefore, the quantity at which the firm maximizes its profit is:
[tex]\[
\boxed{3}
\][/tex]
