Certainly! Let’s solve this problem step-by-step to find out which quantity maximizes the profit:
### Step 1: Understand the Data
We are given the following data in the table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Quantity} & \text{Price} & \text{Total Revenue} & \text{Marginal Revenue} & \text{Total Cost} & \text{Marginal Cost} & \text{Profit or Loss} \ (\text{TR - TC}) \\
\hline
1 & \$ 220 & \$ 220 & & \$ 14 & & \\
\hline
2 & \$ 19 & \$ 38 & & \$ 24 & & \\
\hline
3 & \$ 18 & \$ 54 & & \$ 39 & & \\
\hline
4 & \$ 17 & 968 & & 561 & & \\
\hline
5 & 516 & 580 & & \$ 95 & & \\
\hline
\end{array}
\][/tex]
### Step 2: Calculate the Profit or Loss for Each Quantity
To find the profit or loss for each quantity, we need to subtract the Total Cost (TC) from the Total Revenue (TR) for each quantity. According to the given data:
- For Quantity = 1:
[tex]\[
\text{Profit or Loss} = \$ 220 - \$ 14 = \$ 206
\][/tex]
- For Quantity = 2:
[tex]\[
\text{Profit or Loss} = \$ 38 - \$ 24 = \$ 14
\][/tex]
- For Quantity = 3:
[tex]\[
\text{Profit or Loss} = \$ 54 - \$ 39 = \$ 15
\][/tex]
- For Quantity = 4:
[tex]\[
\text{Profit or Loss} = \$ 968 - \$ 561 = \$ 407
\][/tex]
- For Quantity = 5:
[tex]\[
\text{Profit or Loss} = \$ 580 - \$ 95 = \$ 485
\][/tex]
### Step 3: Summarize the Profits or Losses
So, we have the following profit or loss for each quantity:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Quantity} & \text{Profit or Loss} \\
\hline
1 & \$ 206 \\
\hline
2 & \$ 14 \\
\hline
3 & \$ 15 \\
\hline
4 & \$ 407 \\
\hline
5 & \$ 485 \\
\hline
\end{array}
\][/tex]
### Step 4: Determine the Maximum Profit
From the table, we can see that the highest profit occurs at quantity 5 with a profit of \$485.
### Conclusion
The business maximizes its profit at a quantity of 5.
