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

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

Quantity of shirts sold Marginal cost Total cost Marginal benefit

0 \$0 \$500 \$0

25 \$125 \$625 \$250

50 \$250 \$750 \$500

75 \$375 \$875 \$750

100 \$500 \$1,000 \$1,000

125 \$625 \$1,125 \$1,250

A0
B125
C100
D75

The retailer faces a fixed monthly cost of $\$500$ to maintain the online shop. Additionally, each shirt purchased has a marginal cost of $\$5$. If we let $x$ represent the number of shirts sold, the variable cost is $\$5x$. Consequently, the total cost $TC(x)$ is the sum of the fixed and variable costs:
\[TC(x) = 500 + 5x\]
Each shirt provides a marginal benefit (revenue) of $\$10$. Thus, the total benefit $TB(x)$ is:
\[TB(x) = 10x\]
We are looking for the break-even point where the total costs equal the total benefits, which is defined by the equation:
\[TC(x) = TB(x)\]

Solving for the Number of Shirts

To find the number of shirts $x$ that the retailer needs to sell, we set the total cost expression equal to the total benefit expression:
\[500 + 5x = 10x\]
Subtracting $5x$ from both sides of the equation, we get:
\[500 = 10x - 5x\]
\[500 = 5x\]
Dividing both sides by $5$:
\[x = \frac{500}{5} = 100\]
Thus, the retailer must sell $100$ shirts for the total costs to equal the total benefits.

Verification and Conclusion

We verify this result by checking the provided table. For $x = 100$:

Total cost: $TC(100) = 500 + 5(100) = 500 + 500 = \$\text{1,000}$
Total benefit: $TB(100) = 10(100) = \$\text{1,000}$

Since both values are equal at $x = 100$, the calculation is correct. Comparing this result with the given options, it corresponds to option C. Therefore, the retailer needs to sell $\boxed{100}$ shirts.

Examples

Understanding break-even analysis is essential for any business owner. For instance, if you operate a lemonade stand, your fixed costs might include the cost of renting the table, while variable costs include lemons and sugar. By calculating your break-even point—the number of cups you need to sell to cover your total costs—you can set realistic sales goals and ensure your business becomes profitable. This simple algebraic process helps you determine when your revenue begins to exceed your expenses.

Quantity of shirts sold	Marginal cost	Total cost	Marginal benefit
0	\$0	\$500	\$0
25	\$125	\$625	\$250
50	\$250	\$750	\$500
75	\$375	\$875	\$750
100	\$500	\$1,000	\$1,000
125	\$625	\$1,125	\$1,250