Question - Calculating Future Value of an Account with Compound Interest
mathotherfuture value calculationcompound interest formulainterest rate calculationfinancial mathematics
Solution:
To solve for the value of the account after 8 years, we need to use the given formula for A(t), where t represents the time in years. The initial amount in the account is $5280, the interest rate is 4.2% (or 0.042 as a decimal), and it's compounded monthly (indicated by 12 in the formula).The formula given is:\[ A(t) = P \left(1 + \frac{r}{n}\right)^{nt} \]Where:- $$ A(t) $$ is the amount of money accumulated after t years, including interest.- $$ P $$ is the principal amount (the initial amount of money).- $$ r $$ is the annual interest rate (decimal).- $$ n $$ is the number of times that interest is compounded per year.- $$ t $$ is the time the money is invested for, in years.Given values:- $$ P = \$5280 $$- $$ r = 0.042 $$ (which is 4.2% expressed as a decimal)- $$ n = 12 $$ (since interest is compounded monthly)- $$ t = 8 $$ (since we want to know the value after 8 years)Let's plug these values into the formula:\[ A(8) = 5280 \left(1 + \frac{0.042}{12}\right)^{12 \times 8} \]Now calculate the value inside the parentheses first:\[ 1 + \frac{0.042}{12} = 1 + 0.0035 = 1.0035 \]Then raise this value to the power of $$ 12 \times 8 $$:\[ (1.0035)^{96} \] (since $$ 12 \times 8 = 96 $$)Calculating $$ (1.0035)^{96} $$ gives us approximately:\[ (1.0035)^{96} \approx 1.3991 \] (rounded to four decimal places)Now multiply this result by the principal amount to find $$ A(8) $$:\[ A(8) = 5280 \times 1.3991 \approx 7392.61 \]Therefore, the value of the account after 8 years is approximately $7392.61.
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