Sam is planning to start a pool cleaning business. If Sam meets his profit goal of $45,000 the first year and expects his annual profits to increase by 5.5% each year for the next 3 years, how much profit does Sam expect to make in his third year of business?
$A=P(1+r)^2$
A. $47,475.00
B. $50,086.13
C. $52,840.86
D. $55,747.11
Answer (2)
Sam expects to make approximately $52,840.86 in his third year of business, given his initial profit of $45,000 and an annual increase of 5.5%. This was calculated using the compound interest formula. Hence, the answer is option C.
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Define the initial profit, annual increase rate, and number of years.
Apply the compound interest formula: P n = P 0 ( 1 + r ) n .
Substitute the given values: P 0 = 45000 , r = 0.055 , and n = 3 .
Calculate the profit after 3 years and round to two decimal places: 52840.86 .
Explanation
Problem Analysis Let's analyze the problem. Sam's initial profit is $45,000, and it increases by 5.5% each year for 3 years. We need to find the profit in the third year.
Formula Introduction We can use the formula for compound interest to calculate the profit after each year. The formula for the profit after n years is: P n = P 0 ( 1 + r ) n where:
P n is the profit after n years
P 0 is the initial profit
r is the annual increase rate (as a decimal)
n is the number of years
Applying the Formula In this case, P 0 = 45000 , r = 0.055 , and n = 3 . So, we need to calculate P 3 :
P 3 = 45000 ( 1 + 0.055 ) 3
Calculation Now, let's calculate the value: P 3 = 45000 ( 1.055 ) 3 P 3 = 45000 × 1.174241375 P 3 = 52840.861875 Rounding to two decimal places, we get: P 3 = 52840.86
Final Answer Therefore, Sam's expected profit in the third year is $52,840.86.
Examples
Understanding compound growth is useful in many real-life scenarios. For example, when you invest money in a savings account or a retirement fund, the interest earned compounds over time, similar to Sam's profit growth. This compounding effect can significantly increase your investment's value over the long term. Also, businesses use similar calculations to project revenue growth, plan budgets, and make strategic decisions.
