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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
\begin{tabular}{c}
Quantity of shirts \\
sold
\end{tabular} & \multicolumn{1}{|c|}{\begin{tabular}{c}
Marginal \\
cost
\end{tabular}} & \multicolumn{1}{|c|}{\begin{tabular}{c}
Total \\
cost
\end{tabular}} & \multicolumn{1}{|c|}{\begin{tabular}{c}
Marginal \\
benefit
\end{tabular}} \\
\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]$1,250$[/tex] \\
\hline
\end{tabular}

A. 25

B. 75

C. 100

D. 125

Sagot :

GinnyAnswer
To determine the minimum number of shirts the retailer needs to sell in order to cover all its costs in a month, let's break down the given information step-by-step.

1. Marginal Cost and Benefit per Shirt:
- Marginal cost per shirt: [tex]\( \$5 \)[/tex]
- Marginal benefit per shirt: [tex]\( \$10 \)[/tex]

2. Total Cost:
- There is a fixed total cost of [tex]\( \$500 \)[/tex].

3. Objective:
- We need to find the minimum number of shirts sold such that the total benefit (revenue) covers or exceeds the total cost.

4. Total Benefit Calculation:
The total benefit can be calculated by multiplying the number of shirts sold by the marginal benefit per shirt.

5. Comparison:
We will compare the total cost with the total benefit to find the breakeven point.

Let's go through each quantity in the table to see where the total benefit meets or exceeds the total cost:

- When 0 shirts are sold:
- Total benefit [tex]\(= 0 \times 10 = \$0\)[/tex]
- Total cost [tex]\(= \$500\)[/tex]

The total benefit (\[tex]$0) does not cover the total cost (\$[/tex]500).

- When 25 shirts are sold:
- Total benefit [tex]\(= 25 \times 10 = \$250\)[/tex]
- Total cost [tex]\(= \$625\)[/tex]

The total benefit (\[tex]$250) does not cover the total cost (\$[/tex]500).

- When 50 shirts are sold:
- Total benefit [tex]\(= 50 \times 10 = \$500\)[/tex]
- Total cost [tex]\(= \$750\)[/tex]

The total benefit (\[tex]$500) covers the total cost (\$[/tex]500).

- When 75 shirts are sold:
- Total benefit [tex]\(= 75 \times 10 = \$750\)[/tex]
- Total cost [tex]\(= \$875\)[/tex]

The total benefit (\[tex]$750) exceeds the total cost (\$[/tex]500), but we already found that this happens when 50 shirts are sold for the first time.

Therefore, the minimum number of shirts the retailer needs to sell to cover their costs is [tex]\(\boxed{50}\)[/tex].
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