Question - Calculating Future Value of an Account with Compound Interest
mathfuture value calculationcompound interest formulaprincipal amount and interest rateaccount value after 8 years
Solution:
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.
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