How much would you have to deposit in an account with a $7.5 \%$ interest rate, compounded continuously, to have $ \$2500$ in your account 8 years later?
$P=\$[?]$
Answer (2)
Use the continuous compounding formula: A = P e r t .
Plug in the given values: 2500 = P e 0.075 × 8 .
Solve for P: P = e 0.075 × 8 2500 .
Calculate P: P ≈ $1372.03 .
Explanation
Understanding the Problem We are given that the future value of an account is $2500, the interest rate is 7.5% compounded continuously, and the time period is 8 years. We want to find the principal amount, P, that needs to be deposited to achieve this future value.
Recalling the Formula The formula for continuous compounding is: A = P e r t where:
A is the future value
P is the principal (the initial deposit)
r is the interest rate (as a decimal)
t is the time in years
Identifying Given Values We are given:
A = $2500
r = 7.5% = 0.075
t = 8 years We want to find P .
Substituting Values and Solving for P Plug the given values into the formula: 2500 = P e 0.075 × 8 Now, solve for P :
P = e 0.075 × 8 2500 P = e 0.6 2500
Calculating the Principal Calculating the value of P :
P ≈ 1.8221188 2500 ≈ 1372.03 Therefore, you would need to deposit approximately $1372.03 to have $2500 in your account after 8 years.
Examples
Continuous compounding is a concept used in finance to model investments that grow constantly over time. For example, if you invest in a bond fund that yields a continuously compounded interest rate, this formula helps you calculate how much you need to invest today to reach a specific financial goal in the future, such as retirement savings. Understanding continuous compounding allows for precise financial planning and forecasting.
To have $2500 in an account after 8 years at a 7.5% interest rate compounded continuously, you need to deposit approximately $1372.03. This is calculated using the formula for continuous compounding. By substituting the values into the formula and solving for P, we arrive at the required deposit amount.
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