### Detailed Solution for Part 2 Question 2
To find out how much the business owner will have saved after 7 years by making annual profits deposited at an annual return of 9%, starting with an initial deposit of [tex]$22,000 made today, let's break down the problem into two primary components:
1. Value of the annuity (annual deposits): Calculation of the future value of a series of equal annual deposits made at the end of each year.
2. Value of the initial deposit: Calculation of the future value of the initial deposit made at the start of the investment period.
#### Component 1: Future Value of the Annuity
The future value of an annuity formula is given by:
\[ FV_{\text{annuity}} = P \times \left(\frac{(1 + r)^n - 1}{r}\right) \]
Where:
- \( P \) = annual deposit ($[/tex]22,000 in this case)
- [tex]\( r \)[/tex] = annual return rate (0.09 in this case)
- [tex]\( n \)[/tex] = number of years (7 years in this case)
The future value of the annuity at the end of the period (considering they are made at the end of each year but compounded annually) also needs to include one more year of growth because the first deposit is made at the start of the investment period.
#### Component 2: Future Value of the Initial Deposit
The future value of the initial deposit is given by:
[tex]\[ FV_{\text{initial}} = P \times (1 + r)^n \][/tex]
Where:
- [tex]\( P \)[/tex] = initial deposit ([tex]$22,000 in this case)
- \( r \) = annual return rate (0.09 in this case)
- \( n \) = number of years (7 years in this case)
#### Solution:
Now, let's calculate both components:
1. Future Value of the Annuity (Annual Deposits):
\[ FV_{\text{annuity}} = 22000 \times \left(\frac{(1 + 0.09)^7 - 1}{0.09}\right) \times (1 + 0.09) = 220626.42 \, \text{(approximately)} \]
2. Future Value of the Initial Deposit:
\[ FV_{\text{initial}} = 22000 \times (1 + 0.09)^7 = 40216.86 \, \text{(approximately)}\]
Combining both:
\[ \text{Total Amount Saved} = FV_{\text{annuity}} + FV_{\text{initial}} = 220626.42 + 40216.86 = 260843.28 \, \text{(approximately)} \]
Thus, the total amount saved by the business owner after 7 years would be approximately $[/tex]260,843.28.
