How much would you have to deposit in an account with a $4.75 \%$ interest rate, compounded continuously, to have $ \$ 20,000$ in your account 20 years later? $P=\$[?]$ Round to the nearest cent.
Answer (2)
To have $20,000 in an account after 20 years at a 4.75% interest rate compounded continuously, you need to deposit approximately $7,734.82 today. This is calculated using the formula for continuous compounding. By substituting the values and solving, we find the required present value.
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Use the continuous compounding formula: A = P e r t .
Plug in the given values: 20000 = P e 0.0475 × 20 .
Solve for P: P = e 0.0475 × 20 20000 .
Calculate the value of P and round to the nearest cent: P = $7734.82 .
Explanation
Understanding the Problem We are given the future value of an account, the interest rate, and the time period. We need to find the present value (principal) that needs to be deposited in the account to reach the given future value. The interest is compounded continuously, so we will use the formula for continuous compounding.
Stating the Formula The formula for continuous compounding is: A = P e r t where:
A is the future value
P is the principal (present value)
r is the interest rate (as a decimal)
t is the time in years
Identifying the Given Values We are given:
A = $20,000
r = 4.75% = 0.0475
t = 20 years We need to find P.
Substituting the Values and Solving for P Plug the given values into the formula: 20000 = P e 0.0475 × 20 Now, solve for P: P = e 0.0475 × 20 20000 P = f r a c 20000 e 0.95
Calculating the Present Value Calculate the value of P: P = f r a c 20000 e 0.95 ≈ f r a c 20000 2.585705795 ≈ 7734.82 Round to the nearest cent: P ≈ $7734.82
Examples
Understanding compound interest is crucial for financial planning. For instance, if you want to save for your child's college education, knowing how much to deposit today to reach a specific goal in the future is essential. This problem demonstrates how to calculate the initial investment needed, considering the power of continuous compounding over time. This concept is also applicable in calculating loan payments or understanding the growth of investments in various financial instruments.
