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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 costTotal costMarginal 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





A75
B0
C125
D100

Verified Answer :

  • Establish the cost and revenue functions as \(C(q) = 500 + 5q\) and \(B(q) = 10q\).
  • Solve the equation \(500 + 5q = 10q\) to find the break-even point.
  • Verify the solution \(q = 100\) using the provided data table, confirming that total costs equal total benefits at this quantity.
  • The retailer needs to sell \(\boxed{100}\) shirts.

Explanation

  1. Defining Total Cost and Total Benefit Functions

To determine the number of shirts the retailer needs to sell for its total costs to be equal to its total benefits, we begin by defining the financial functions involved. The fixed cost for operating the online shop is \(\$500\) per month. Each shirt costs \(\$5\) to acquire, which is the marginal cost. If we let \(q\) represent the number of shirts sold, the total cost \(C(q)\) can be represented by the linear function:
\[C(q) = 500 + 5q\]
The retailer sells each shirt for \(\$10\), which is the marginal benefit per unit. Thus, the total benefit \(B(q)\) earned from selling \(q\) shirts is:
\[B(q) = 10q\]

  1. Solving for the Break-Even Quantity

To find the break-even point where the retailer's total costs equal their total benefits, we set the two functions equal to each other:
\[C(q) = B(q)\]
Substituting the expressions defined in the previous step:
\[500 + 5q = 10q\]
Now, we solve for \(q\). First, subtract \(5q\) from both sides of the equation:
\[500 = 10q - 5q\]
\[500 = 5q\]
Finally, divide both sides by \(5\):
\[q = \frac{500}{5} = 100\]
This calculation shows that the retailer must sell \(100\) shirts for the total costs to equal the total benefits.

  1. Verification with the Data Table

We can verify our result using the provided data table. The table shows the total cost and total benefit for different quantities of shirts sold. Looking at the row for 100 shirts sold, the total cost is \(\$1,000\) and the total benefit is \(\$1,000\). Since both values are equal, this confirms our algebraic finding that the break-even point is \(100\) shirts. Thus, the correct option is D. The final answer is \(\boxed{100}\).

Examples

Understanding break-even analysis is essential for any business owner, from a small lemonade stand to a large corporation. By calculating exactly how many units must be sold to cover all expenses, an owner can set realistic sales goals and determine the viability of a business model. This simple algebraic process helps ensure that a company can achieve profitability by identifying the exact point where revenue surpasses initial costs.

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