Let's break down the problem into parts, step-by-step.
### Part (a)
Calculating the amount in the account after 15 years:
To find out how much money there will be in the account after 15 years, we use the formula for the future value of a series of equal payments compounded at regular intervals. This formula is:
[tex]\[ FV = P \times \frac{(1 + r)^n - 1}{r} \][/tex]
Where:
- [tex]\( P \)[/tex] is the monthly deposit
- [tex]\( r \)[/tex] is the monthly interest rate
- [tex]\( n \)[/tex] is the total number of payments
Given values:
- Monthly deposit ( [tex]\( P \)[/tex] ) = [tex]$500
- Annual interest rate = 4%, so the monthly interest rate (\( r \)) = 4% / 12 = 0.04 / 12 ≈ 0.003333
- Number of years = 15, thus the number of monthly payments (\( n \)) = 15 * 12 = 180
Using these values, you can calculate the future value (FV):
The future value after 15 years is:
\[ FV = 500 \times \frac{(1 + 0.003333)^{180} - 1}{0.003333} \]
This calculates to approximately $[/tex]123,045.24.
### Part (b)
Calculating the total money deposited:
The total money deposited is the amount of the monthly deposit multiplied by the number of months.
Total money deposited (Total Deposits) = Monthly deposit [tex]\(\times\)[/tex] Number of months
Given:
- Monthly deposit = [tex]$500
- Number of months = 15 years \(\times\) 12 months/year = 180 months
So,
\[ \text{Total Deposits} = 500 \times 180 \]
Total money deposited equals $[/tex]90,000.
### Part (c)
Calculating the total interest earned:
Total interest earned is the difference between the future value of the account and the total money deposited.
Total interest earned (Total Interest) = Future Value [tex]\( - \)[/tex] Total Deposits
Using the values obtained:
- Future Value = [tex]$123,045.24
- Total Deposits = $[/tex]90,000
So,
[tex]\[ \text{Total Interest} = 123,045.24 - 90,000 \][/tex]
Total interest earned equals approximately [tex]$33,045.24.
### Summary:
a) The amount in the account after 15 years will be approximately $[/tex]123,045.24.
b) The total money deposited over the 15 years will be [tex]$90,000.
c) The total interest earned over the 15 years will be approximately $[/tex]33,045.24.
