Calculating Future Value of an Account with Compound Interest
The image shows a mathematical formula for the future value of an account and asks for the value of the account after 8 years. The formula provided is a standard compound interest formula:
\[ A(t) = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A(t) \) is the amount of money accumulated after time \( t \), including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (in decimal form).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested for, in years.
Given in the problem:
- \( P = 5280 \) (the principal amount)
- \( r = 0.042 \) (the annual interest rate, in decimal form)
- \( n = 12 \) (since interest is compounded monthly, there are 12 compounding periods per year)
- \( t = 8 \) (the time period in question, namely 8 years)
Plugging these values into the formula, we get:
\[ A(8) = 5280 \left(1 + \frac{0.042}{12}\right)^{12 \times 8} \]
Calculating each part step by step:
First, divide the annual interest rate by the number of compounding periods:
\[ \frac{r}{n} = \frac{0.042}{12} = 0.0035 \]
Then add 1 to this result:
\[ 1 + \frac{r}{n} = 1 + 0.0035 = 1.0035 \]
Calculate the exponent, which is the number of compounding periods times the number of years:
\[ nt = 12 \times 8 = 96 \]
Now raise the previous result to this exponent:
\[ \left(1 + \frac{r}{n}\right)^{nt} = 1.0035^{96} \]
And finally, multiply this by the principal amount:
\[ A(8) = 5280 \times 1.0035^{96} \]
Using a calculator, you can compute:
\[ 1.0035^{96} \approx 1.368616 \]
Multiplying this by the principal amount, we get:
\[ A(8) = 5280 \times 1.368616 \approx 7226.933 \]
So the value of the account after 8 years is approximately $7226.93.
