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

How many shirts would the retailer need to sell for its total costs to be equal to its marginal benefits?

\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 & \$1,250 \\
\hline
\end{tabular}



Answer :

GinnyAnswer
Let's solve the problem step-by-step:

1. Understand the given values:
- Fixed cost to keep the online shop active: [tex]\( \$ 500 \)[/tex] per month.
- Marginal cost (cost per additional shirt produced/purchased): [tex]\( \$ 5 \)[/tex] per shirt.
- Marginal benefit (revenue per additional shirt sold): [tex]\( \$ 10 \)[/tex] per shirt.

2. Identify the goal:
- We want to find the quantity of shirts sold where the total cost is equal to the total benefit.

3. Define the relationships:
- Total cost is given by [tex]\( \text{Total Cost} = \text{Fixed Cost} + \text{Marginal Cost per Shirt} \times \text{Quantity Sold} \)[/tex].
- Total benefit is given by [tex]\( \text{Total Benefit} = \text{Marginal Benefit per Shirt} \times \text{Quantity Sold} \)[/tex].

4. Set up the equations:
- We have a fixed cost of [tex]\( \$ 500 \)[/tex].
- We have a marginal cost of [tex]\( \$ 5 \)[/tex] per shirt.
- We have a marginal benefit of [tex]\( \$ 10 \)[/tex] per shirt.

Therefore:
[tex]\[ \text{Total Cost} = 500 + 5 \times Q \][/tex]
[tex]\[ \text{Total Benefit} = 10 \times Q \][/tex]

5. Set the total cost equal to the total benefit:
[tex]\[ 500 + 5Q = 10Q \][/tex]

6. Solve for [tex]\( Q \)[/tex] (Quantity of shirts):
- Subtract [tex]\( 5Q \)[/tex] from both sides:
[tex]\[ 500 = 5Q \][/tex]
- Divide both sides by 5:
[tex]\[ Q = 100 \][/tex]

7. Verify the solution:
- When [tex]\( Q = 100 \)[/tex]:
[tex]\[ \text{Total Cost} = 500 + 5 \times 100 = 1000 \][/tex]
[tex]\[ \text{Total Benefit} = 10 \times 100 = 1000 \][/tex]

So, the retailer needs to sell 100 shirts for the total costs to be equal to the total benefits. This means at [tex]\( Q = 100 \)[/tex] shirts, both the total cost and total benefit are [tex]\( \$ 1000 \)[/tex].

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