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Sam is planning to start a pool cleaning business. If Sam meets his profit goal of [tex]$\$[/tex]45,000[tex]$ the first year and expects his annual profits to increase by $[/tex]5.5\%$ each year for the next 3 years, how much profit does Sam expect to make in his third year of business?

The formula for compound interest is [tex]$\quad A = P(1 + r)^2$[/tex].

a. [tex]$\$[/tex]47,475.00$
b. [tex]$\$[/tex]50,086.13$
c. [tex]$\$[/tex]52,840.86$
d. [tex]$\$[/tex]55,747.11$

Please select the best answer from the choices provided:
A
B
C
D



Answer :

Sam is planning to start a pool cleaning business with an initial profit goal of $45,000 the first year. The annual profit is expected to increase by \(5.5\%\).

To calculate the profit for the third year, we can use the formula for compound interest which is:

[tex]\[ A = P(1 + r)^n \][/tex]

Here:
- \( P \) is the initial amount (initial profit), which is $45,000.
- \( r \) is the annual increase rate, which is \(5.5\%\) or \(0.055\) as a decimal.
- \( n \) is the number of years, which is 2 years (from the first year to the third year).

Plug these values into the formula:

[tex]\[ A = 45000 \times (1 + 0.055)^2 \][/tex]

First, calculate \(1 + 0.055\):

[tex]\[ 1 + 0.055 = 1.055 \][/tex]

Next, raise this value to the power of 2:

[tex]\[ 1.055^2 \approx 1.113 \][/tex]

Finally, multiply this by the initial profit:

[tex]\[ A = 45000 \times 1.113 \approx 50086.13 \][/tex]

Therefore, the expected profit for the third year is approximately $50,086.13.

The best answer from the choices provided is:

b. \$50,086.13