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 $20,000 for 6 years at an interest rate of 6.5% if the money is
a. compounded semiannually;
b. compounded quarterly;
c. compounded continuously.
a. What is the accumulated value if the money is compounded semiannually?
b. What is the accumulated value if the money is compounded quarterly?
c. What is the accumulated value if the money is compounded monthly?
d. What is the accumulated value if the money is compounded continuously?
Answer (2)
Identify the principal amount P = $20 , 000 , the interest rate r = 0.065 , and the time t = 6 years.
Apply the formula for continuous compounding: A = P e r t .
Substitute the given values into the formula: A = 20000 × e 0.065 × 6 .
Calculate the accumulated value and round to the nearest cent: A = $29539.62 .
Explanation
Understanding the Problem We are asked to find the accumulated value of an investment of $20,000 for 6 years at an interest rate of 6.5% compounded continuously. We will use the formula for continuous compounding, which is given by: A = P e r t where:
A is the accumulated value
P is the principal amount
r is the interest rate (as a decimal)
t is the time in years
Substituting the Values We are given:
P = $20 , 000
r = 6.5% = 0.065
t = 6 years We substitute these values into the formula: A = 20000 × e 0.065 × 6 A = 20000 × e 0.39
Calculating the Accumulated Value Now, we calculate e 0.39 . The result of this operation is approximately 1.477. A = 20000 × 1.477 A = 29540
Final Answer Therefore, the accumulated value of the investment when compounded continuously is approximately $29540. We need to round the answer to the nearest cent, so we have: A = $29539.62
Examples
Imagine you deposit money into a bank account that compounds interest continuously. This concept is used to calculate how much money you'll have after a certain period, considering the principal amount, interest rate, and time. Continuous compounding provides the highest yield compared to other compounding frequencies (e.g., annually, quarterly, or monthly). Understanding continuous compounding helps you make informed decisions about investments and savings, allowing you to estimate the future value of your money accurately. This is particularly useful in long-term financial planning, such as retirement savings or investment portfolios.
The accumulated values for an investment of $20,000 at an interest rate of 6.5% are as follows: Compounded Semiannually: $28,515.38; Quarterly: $28,615.48; Monthly: $28,688.34; and Continuously: $29,542.42.
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