Let's carefully follow the steps required to determine how many shirts the retailer needs to sell for its total costs to equal its total benefits.
1. Understand the Components:
- Monthly fixed cost: \[tex]$500
- Marginal cost per shirt: \$[/tex]5
- Marginal benefit per shirt: \[tex]$10
2. Define Total Cost and Total Benefit:
- Total Cost (TC): This includes the monthly fixed cost plus the cost of producing a certain number of shirts.
- Total Benefit (TB): This is the revenue generated from selling a certain number of shirts.
3. Formulate The Equations:
- Total Cost (TC):
\[
\text{TC} = 500 + 5 \times \text{quantity}
\]
- Total Benefit (TB):
\[
\text{TB} = 10 \times \text{quantity}
\]
We want to find the quantity where these two are equal:
\[
\text{TC} = \text{TB}
\]
4. Set Up The Equation:
\[
500 + 5 \times \text{quantity} = 10 \times \text{quantity}
\]
5. Solve for Quantity:
- First, we will bring all terms involving quantity to one side of the equation:
\[
500 = 10 \times \text{quantity} - 5 \times \text{quantity}
\]
- Simplify:
\[
500 = 5 \times \text{quantity}
\]
- Solve for the quantity:
\[
\text{quantity} = \frac{500}{5} = 100
\]
Thus, the retailer would need to sell 100 shirts to make the total cost equal to the total benefit.
By looking at the provided table for verification:
\begin{tabular}{|l|l|l|l|}
\hline Quantity of shirts sold & Marginal cost & Total cost & Marginal benefit \\
\hline 0 & \$[/tex]0 & \[tex]$500 & \$[/tex]0 \\
\hline 25 & \[tex]$125 & \$[/tex]625 & \[tex]$250 \\
\hline 50 & \$[/tex]250 & \[tex]$750 & \$[/tex]500 \\
\hline 75 & \[tex]$375 & \$[/tex]875 & \[tex]$750 \\
\hline 100 & \$[/tex]500 & \[tex]$1,000 & \$[/tex]1,000 \\
\hline 125 & \[tex]$625 & \$[/tex]1,125 & \[tex]$1,250 \\
\hline
\end{tabular}
We see that when the quantity is 100, both the total cost (\$[/tex]1,000) and the total benefit (\$1,000) are equal.
Therefore, the correct answer is:
B. 100
