Korey is planning to open a comic book store. His store will cost $12,500 to open and will come with an $8,000 total annual operational cost. If his store makes $12,000 in profit during the first year, and profits increase by 6% each year from then, how long will it take for Korey to see an overall profit in his business (where total profits exceed total expenses)?
[tex]A=P(1+r)^t[/tex]
A. Korey will see an overall profit in his first year of business.
B. Korey will see an overall profit in his second year of business.
C. Korey will see an overall profit in his third year of business.
D. Korey will see an overall profit in his fourth year of business.
Answer (2)
Korey will see an overall profit in his third year of business after his cumulative profits exceed his total expenses. After two years of losses, by the end of the third year, he will achieve a profit of $1,703.20. The answer is C.
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Calculate the cumulative profit for each year, considering the 6% annual increase: Year 1 profit is $12 , 000 , Year 2 profit is $12 , 720 , and Year 3 profit is $13 , 483.20 .
Calculate the total expenses for each year, including the initial cost of $12 , 500 and an annual operational cost of $8 , 000 .
Compare the cumulative profit with the total expenses for each year.
Determine that Korey will see an overall profit in his third year of business, as the cumulative profit exceeds the total expenses after three years. $\boxed{c}
Explanation
Problem Analysis Let's analyze Korey's comic book store finances to determine when his overall profits will exceed his total expenses. We need to consider the initial costs, annual operational costs, and the increasing annual profits.
Defining Costs and Profits The initial cost to open the store is $12 , 500 , and the annual operational cost is $8 , 000 . The profit in the first year is $12 , 000 , and it increases by 6% each year. We need to find the year when the cumulative profit exceeds the total expenses.
Calculating Cumulative Profit and Total Expenses Let's calculate the cumulative profit and total expenses for each year:
Year 1:
Profit: $12 , 000
Cumulative Profit: $12 , 000
Total Expenses: $12 , 500 + $8 , 000 = $20 , 500
Year 2:
Profit: $12 , 000 × 1.06 = $12 , 720
Cumulative Profit: $12 , 000 + $12 , 720 = $24 , 720
Total Expenses: $12 , 500 + ( $8 , 000 × 2 ) = $28 , 500
Year 3:
Profit: $12 , 720 × 1.06 = $13 , 483.20
Cumulative Profit: $24 , 720 + $13 , 483.20 = $38 , 203.20
Total Expenses: $12 , 500 + ( $8 , 000 × 3 ) = $36 , 500
After 3 years, the cumulative profit ( $38 , 203.20 ) exceeds the total expenses ( $36 , 500 ).
Conclusion Therefore, Korey will see an overall profit in his third year of business.
Examples
Understanding when a business becomes profitable is crucial for entrepreneurs. This problem demonstrates how to calculate cumulative profits and total expenses over time, considering factors like initial investment, operational costs, and profit growth. For instance, if you're starting a small bakery, you need to know when your total earnings will cover your initial equipment costs, ingredient expenses, and other overheads. By projecting your sales and costs, you can estimate the time it will take to break even and start making a profit, helping you make informed decisions about pricing, production, and investments.
