To determine the minimum number of shirts the retailer needs to sell in order to cover all its costs for the month, we need to find the break-even point where the total revenue from selling shirts is at least equal to the total costs (which include both fixed and variable costs).
Here's a step-by-step breakdown:
1. Identify the monthly fixed cost:
The retailer incurs a fixed cost of \[tex]$500 per month to keep the online shop active and updated.
2. Identify the marginal (variable) costs and benefits:
- The marginal cost of acquiring each shirt is \$[/tex]5 per shirt.
- The marginal benefit or selling price of each shirt is \$10 per shirt.
3. Set up the break-even condition:
Let [tex]\( x \)[/tex] be the number of shirts sold in a month. The total cost per month includes both fixed costs and marginal costs:
[tex]\[
\text{Total Cost} = \text{Fixed Cost} + (\text{Marginal Cost per Shirt} \times x)
\][/tex]
The total revenue is:
[tex]\[
\text{Total Revenue} = \text{Marginal Benefit per Shirt} \times x
\][/tex]
4. Formulate the break-even equation:
The retailer breaks even when total revenue equals total costs:
[tex]\[
\text{Fixed Cost} + (\text{Marginal Cost per Shirt} \times x) \leq \text{Marginal Benefit per Shirt} \times x
\][/tex]
Substituting the values we have:
[tex]\[
500 + 5x \leq 10x
\][/tex]
5. Solve the inequality:
[tex]\[
500 + 5x \leq 10x
\][/tex]
Subtract [tex]\( 5x \)[/tex] from both sides:
[tex]\[
500 \leq 5x
\][/tex]
Divide by 5:
[tex]\[
100 \leq x
\][/tex]
Thus, the retailer needs to sell at least [tex]\( x = 100 \)[/tex] shirts to cover all monthly costs.
So the minimum number of shirts the retailer needs to sell to pay for all its costs in a month is:
D. 100
