Answer :
To solve this question, let's break it down step by step.
1. Calculate the annual profit for the first year:
Korey's expected monthly profit is \$1,000. Since there are 12 months in a year, his annual profit in the first year would be:
[tex]\[ \text{Annual Profit in First Year} = 1000 \times 12 = 12000 \text{ dollars} \][/tex]
2. Determine the annual growth rate:
Korey expects his profits to increase by 6% each year. The growth rate \( r \) can be expressed as a decimal:
[tex]\[ r = 0.06 \][/tex]
3. Calculate the profits for the fifth year:
We have the formula for compound interest:
[tex]\[ A = P (1 + r)^n \][/tex]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (initial profit).
- \( r \) is the annual growth rate.
- \( n \) is the number of years.
Here, \( P \) is \$12,000, \( r \) is 0.06, and \( n \) is 4 years (since we are calculating for the fifth year of operation, which is four years after the first year).
Plugging in the numbers:
[tex]\[ \text{Profit in Fifth Year} = 12000 \times (1 + 0.06)^4 \][/tex]
Evaluating the expression within the parentheses first:
[tex]\[ 1 + 0.06 = 1.06 \][/tex]
Now raising this to the power of 4:
[tex]\[ 1.06^4 \approx 1.262477 \][/tex]
Finally, multiplying this result by the initial annual profit:
[tex]\[ \text{Profit in Fifth Year} = 12000 \times 1.262477 \approx 15149.72352 \][/tex]
So, the amount Korey expects to make in profits in his fifth year of operation is approximately \$15,149.72.
Therefore, the best answer from the provided choices is:
a. \$15,149.72
1. Calculate the annual profit for the first year:
Korey's expected monthly profit is \$1,000. Since there are 12 months in a year, his annual profit in the first year would be:
[tex]\[ \text{Annual Profit in First Year} = 1000 \times 12 = 12000 \text{ dollars} \][/tex]
2. Determine the annual growth rate:
Korey expects his profits to increase by 6% each year. The growth rate \( r \) can be expressed as a decimal:
[tex]\[ r = 0.06 \][/tex]
3. Calculate the profits for the fifth year:
We have the formula for compound interest:
[tex]\[ A = P (1 + r)^n \][/tex]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (initial profit).
- \( r \) is the annual growth rate.
- \( n \) is the number of years.
Here, \( P \) is \$12,000, \( r \) is 0.06, and \( n \) is 4 years (since we are calculating for the fifth year of operation, which is four years after the first year).
Plugging in the numbers:
[tex]\[ \text{Profit in Fifth Year} = 12000 \times (1 + 0.06)^4 \][/tex]
Evaluating the expression within the parentheses first:
[tex]\[ 1 + 0.06 = 1.06 \][/tex]
Now raising this to the power of 4:
[tex]\[ 1.06^4 \approx 1.262477 \][/tex]
Finally, multiplying this result by the initial annual profit:
[tex]\[ \text{Profit in Fifth Year} = 12000 \times 1.262477 \approx 15149.72352 \][/tex]
So, the amount Korey expects to make in profits in his fifth year of operation is approximately \$15,149.72.
Therefore, the best answer from the provided choices is:
a. \$15,149.72
