How much money must be in an account with an 8% interest rate, compounded continuously, to have $3000 in your account 10 years later?
P = $ [?]
Answer (1)
Use the formula for continuous compounding: A = P e r t .
Plug in the given values: A = 3000 , r = 0.08 , and t = 10 .
Solve for P : P = e 0.08 × 10 3000 .
Calculate the value of P : P ≈ 1347.99 .
Explanation
Understanding the Problem We are given an account with a 8% interest rate, compounded continuously. We want to find the principal amount, P , needed to have $3000 in the account 10 years later.
Recalling the Formula The formula for continuous compounding is given by: A = P e r t where:
A is the future value of the investment/loan, including interest
P is the principal investment amount (the initial deposit or loan amount)
r is the annual interest rate (as a decimal)
t is the time the money is invested or borrowed for, in years
Identifying Given Values We are given:
A = $3000
r = 8% = 0.08
t = 10 years We need to find P .
Solving for P Plugging the given values into the formula, we get: 3000 = P e 0.08 × 10 Now, we solve for P :
P = e 0.08 × 10 3000 P = e 0.8 3000 Using a calculator, we find that: P ≈ 1347.99
Final Answer Therefore, the principal amount needed is approximately $1347.99 .
Examples
Continuous compounding is a concept widely used in finance. For example, if you want to know how much money you need to invest today to reach a specific financial goal in the future, such as retirement savings, you can use the continuous compounding formula. This helps in financial planning and investment decisions, allowing individuals to estimate the initial investment required to achieve their desired future value, considering the effects of continuous interest accumulation over time. Understanding this concept is crucial for making informed decisions about savings, investments, and loans.
