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A retailer spends [tex]$\$[/tex]500[tex]$ per month to keep its online shop active and updated. The store acquires shirts at a marginal cost of $[/tex]\[tex]$5$[/tex] per shirt. Each shirt sells for a marginal benefit of [tex]$\$[/tex]10[tex]$ per shirt.

What is the minimum number of shirts the retailer needs to sell in order to pay for all its costs in a month?

\begin{tabular}{|l|l|l|l|}
\hline
Quantity of shirts sold & Marginal cost & Total cost & Marginal benefit \\
\hline
0 & $[/tex]\[tex]$0$[/tex] & [tex]$\$[/tex]500[tex]$ & $[/tex]\[tex]$0$[/tex] \\
\hline
25 & [tex]$\$[/tex]125[tex]$ & $[/tex]\[tex]$625$[/tex] & [tex]$\$[/tex]250[tex]$ \\
\hline
50 & $[/tex]\[tex]$250$[/tex] & [tex]$\$[/tex]750[tex]$ & $[/tex]\[tex]$500$[/tex] \\
\hline
75 & [tex]$\$[/tex]375[tex]$ & $[/tex]\[tex]$875$[/tex] & [tex]$\$[/tex]750[tex]$ \\
\hline
100 & $[/tex]\[tex]$500$[/tex] & [tex]$\$[/tex]1,000[tex]$ & $[/tex]\[tex]$1,000$[/tex] \\
\hline
125 & [tex]$\$[/tex]625[tex]$ & $[/tex]\[tex]$1,125$[/tex] & [tex]$\$[/tex]1,250$ \\
\hline
\end{tabular}

A. 125
B. 75
C. 25
D. 100



Answer :

GinnyAnswer
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