To determine how many shirts the retailer would need to sell for its total costs to be equal to its marginal benefits, we need to find the point at which the total cost equals the total benefit.
Let's break down the problem step-by-step:
1. Identify the fixed costs and variable costs:
- The retailer has a fixed monthly expenditure of \[tex]$500 to keep the shop active and updated.
- Each shirt incurs a marginal cost of \$[/tex]5.
2. Identify the marginal benefit:
- Each shirt sold provides a marginal benefit of \$10.
3. Formulate the total cost and total benefit equations:
- Total Cost (TC) = Fixed Cost + (Marginal Cost per shirt Quantity of shirts)
- Total Benefit (TB) = Marginal Benefit per shirt Quantity of shirts
4. Set up the equation where Total Cost equals Total Benefit:
- [tex]\(500 + 5Q = 10Q\)[/tex]
5. Solve for the quantity of shirts (Q):
- Subtract [tex]\(5Q\)[/tex] from both sides:
[tex]\[500 = 5Q\][/tex]
- Divide both sides by 5:
[tex]\[Q = \frac{500}{5} = 100\][/tex]
6. Interpret the result:
The retailer needs to sell 100 shirts for the total costs to equal the total benefits.
Therefore, the correct answer is:
[tex]\[
\boxed{100}
\][/tex]
