How much would you have to deposit in an account with an $8 \%$ interest rate, compounded continuously, to have $3000 in your account 10 years later?
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Round to the nearest cent.
Answer (2)
To have $3000 after 10 years with an 8% interest rate compounded continuously, you need to deposit approximately $1347.99. This is calculated using the continuous compounding formula. After solving, we find the required principal amount needed today for that future value.
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Use the continuous compounding formula: A = P e r t .
Plug in the given values: 3000 = P e 0.08 × 10 .
Solve for P: P = e 0.8 3000 .
Calculate P and round to the nearest cent: P = $1347.99 .
Explanation
Understanding the Problem We are given that we want to have $3000 in an account after 10 years with an interest rate of 8% compounded continuously. We need to find the principal amount, P, that needs to be deposited.
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 the Given Values We are given:
A = $3000
r = 8% = 0.08
t = 10 years We need to find P .
Substituting the Values Plugging the given values into the formula, we get: 3000 = P e 0.08 × 10 3000 = P e 0.8 To solve for P , we divide both sides by e 0.8 :
P = e 0.8 3000
Calculating the Principal Calculating the value of P :
P = e 0.8 3000 ≈ 2.22554 3000 ≈ 1347.99 Rounding to the nearest cent, we get $P = $1347.99.
Final Answer Therefore, you would have to deposit $1347.99 in t h e a cco u n tt o ha v e $3000 in 10 years.
Examples
Continuous compounding is a concept widely used in finance. For example, if you want to calculate the present value of a future investment, or if you are analyzing loan payments, understanding continuous compounding helps in making informed financial decisions. It's also used in calculating the growth of populations or decay of radioactive materials, where changes happen continuously over time. By understanding this formula, you can accurately predict the future value of your investments or the amount you need to invest today to reach a specific financial goal in the future. For instance, knowing how much to invest now to have a certain amount for retirement.
