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Sagot :
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]
[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]
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