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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.

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

\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. 100
D. 25



Answer :

GinnyAnswer
To determine how many shirts the retailer needs to sell for its marginal benefits to be greater than its total costs, we need to calculate the costs and benefits for various quantities and compare them.

Let's break down the problem step-by-step:

1. Monthly fixed cost: The retailer has a fixed monthly cost of [tex]$500 to keep the shop active and updated. 2. Marginal cost per shirt: The cost to get each additional shirt is $[/tex]5.

3. Marginal benefit per shirt: The revenue from selling each additional shirt is [tex]$10. We need to find the point where the marginal benefit exceeds the total cost. Let's evaluate the total costs and marginal benefits at different quantities, using the provided table: \[ \text{Total Cost} = \text{Monthly Cost} + \text{Marginal Cost per Shirt} \times \text{Quantity of Shirts Sold} \] \[ \text{Marginal Benefit} = \text{Marginal Benefit per Shirt} \times \text{Quantity of Shirts Sold} \] From the table: - For 0 shirts: - Marginal Cost: $[/tex]0[tex]$ - Total Cost: $[/tex]500 + [tex]$0 = $[/tex]500[tex]$ - Marginal Benefit: $[/tex]10 \times 0 = [tex]$0$[/tex]

- For 25 shirts:
- Marginal Cost: [tex]$5 \times 25 = $[/tex]125[tex]$ - Total Cost: $[/tex]500 + [tex]$125 = $[/tex]625[tex]$ - Marginal Benefit: $[/tex]10 \times 25 = [tex]$250$[/tex]

- For 50 shirts:
- Marginal Cost: [tex]$5 \times 50 = $[/tex]250[tex]$ - Total Cost: $[/tex]500 + [tex]$250 = $[/tex]750[tex]$ - Marginal Benefit: $[/tex]10 \times 50 = [tex]$500$[/tex]

- For 75 shirts:
- Marginal Cost: [tex]$5 \times 75 = $[/tex]375[tex]$ - Total Cost: $[/tex]500 + [tex]$375 = $[/tex]875[tex]$ - Marginal Benefit: $[/tex]10 \times 75 = [tex]$750$[/tex]

- For 100 shirts:
- Marginal Cost: [tex]$5 \times 100 = $[/tex]500[tex]$ - Total Cost: $[/tex]500 + [tex]$500 = $[/tex]1000[tex]$ - Marginal Benefit: $[/tex]10 \times 100 = [tex]$1000$[/tex]

- For 125 shirts:
- Marginal Cost: [tex]$5 \times 125 = $[/tex]625[tex]$ - Total Cost: $[/tex]500 + [tex]$625 = $[/tex]1125[tex]$ - Marginal Benefit: $[/tex]10 \times 125 = [tex]$1250$[/tex]

Now, let's find the point where the marginal benefit exceeds the total cost:

- At 0 shirts: Marginal Benefit ([tex]$0$[/tex]) < Total Cost ([tex]$500$[/tex])
- At 25 shirts: Marginal Benefit ([tex]$250$[/tex]) < Total Cost ([tex]$625$[/tex])
- At 50 shirts: Marginal Benefit ([tex]$500$[/tex]) < Total Cost ([tex]$750$[/tex])
- At 75 shirts: Marginal Benefit ([tex]$750$[/tex]) < Total Cost ([tex]$875$[/tex])
- At 100 shirts: Marginal Benefit ([tex]$1000$[/tex]) = Total Cost ([tex]$1000$[/tex])
- At 125 shirts: Marginal Benefit ([tex]$1250$[/tex]) > Total Cost ([tex]$1125$[/tex])

We can see that at 125 shirts, the marginal benefit ([tex]$1250$[/tex]) is greater than the total cost ([tex]$1125$[/tex]).

Therefore, the retailer needs to sell 125 shirts.

Answer: A. 125

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