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? [tex]A=P(1+r)^{\prime}[/tex]
A. $47,475.00
B. $50,086.13
C. $52,840.86
D. $55,747.11
Answer (1)
Identify the initial profit, annual increase rate, and number of years.
Apply the compound interest formula: A = P ( 1 + r ) t .
Substitute the values: A = 45000 ( 1 + 0.055 ) 3 .
Calculate the profit in the third year: $52 , 840.86 .
Explanation
Understanding the Problem We are given that Sam's initial profit is $45,000 and it increases by 5.5% each year for the next 3 years. We need to find the profit in the third year. The formula for compound interest is given as A = P ( 1 + r ) t , where A is the amount, P is the principal, r is the rate, and t is the time.
Identifying the Variables We identify the values for P, r, and t.
P = 45000 (initial profit)
r = 5.5% = 0.055 (annual increase rate)
t = 3 (number of years)
Applying the Formula Now, we substitute these values into the formula:
A = 45000 ( 1 + 0.055 ) 3
A = 45000 ( 1.055 ) 3
A = 45000 × 1.174241625
A = 52840.8728125
Calculating the Profit Rounding to two decimal places, we get:
A = 52840.86
Therefore, Sam's expected profit in the third year is approximately $52,840.86.
Examples
Understanding compound growth is useful in many real-life situations. For example, when you invest money in a savings account or a retirement fund, the interest earned each year is added to your principal, and the next year's interest is calculated on the new, larger amount. This is compound growth, and it can help your money grow faster over time. Similarly, businesses can use this concept to project their future profits based on a consistent growth rate.
