Use the compound interest formulas [tex]$A = P \left(1+\frac{ r }{ n }\right)^{ nt }$[/tex] and [tex]$A = P e^{ rt }$[/tex] to solve the problem given. Round answers to the nearest cent.
Find the accumulated value of an investment of [tex]$20,000$[/tex] for 6 years at an interest rate of [tex]$6.5 \%$[/tex] if the money is
a. compounded semiannually;
b. compounded continuously.
a. What is the accumulated value if the money is compounded semiannually?
[tex]$ \square$[/tex]
(Round your answer to the nearest cent. Do not include the [tex]$ \$$[/tex] symbol in your answer.)
Answer (2)
Identify the given values: P = 20000 , r = 0.065 , t = 6 , and n = 2 .
Substitute the values into the compound interest formula: A = P ( 1 + n r ) n t .
Calculate A = 20000 × ( 1.0325 ) 12 ≈ 20000 × 1.4678467782217908 .
Round the accumulated value to the nearest cent: A = 29356.94 .
Explanation
Understanding the Problem We are given the principal amount P = $20 , 000 , the time t = 6 years, and the interest rate r = 6.5% = 0.065 . We need to find the accumulated value A when the money is compounded semiannually. The formula for compound interest is given by A = P ( 1 + n r ) n t where P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.
Substituting the Values In this case, we have P = 20000 , r = 0.065 , t = 6 , and n = 2 (since the interest is compounded semiannually). Substituting these values into the formula, we get A = 20000 ( 1 + 2 0.065 ) 2 × 6 A = 20000 ( 1 + 0.0325 ) 12 A = 20000 ( 1.0325 ) 12
Calculating the Accumulated Value Now, we calculate ( 1.0325 ) 12 . The result of this operation is approximately 1.4678467782217908 .
A = 20000 × 1.4678467782217908 A = 29356.935564435815
Final Answer Rounding the accumulated value to the nearest cent, we get A = $29 , 356.94 .
Examples
Compound interest is a powerful tool for growing wealth over time. For example, understanding compound interest can help you plan for retirement. If you invest $10,000 in a retirement account with an average annual return of 7% compounded annually, after 30 years, your investment could grow to approximately $76,122.55. This illustrates how consistent investing and the power of compounding can lead to significant financial gains over the long term.
The accumulated value of a $20,000 investment compounded semiannually at a 6.5% interest rate for 6 years is approximately $28,647.29. This was calculated using the compound interest formula with the appropriate substitutions for principal, rate, time, and compounding frequency.
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