$\$1100$ are deposited in an account with 5.8\% interest rate, compounded continuously. What is the balance after 22 years?
$F=\$[?]
Round to the nearest cent.
Answer (2)
The balance after 22 years in an account with an initial deposit of $1100 and an interest rate of 5.8% compounded continuously is approximately $3940.51. This was calculated using the formula for continuous compounding, substituting the provided values. The final result was rounded to the nearest cent after performing the calculations.
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Identify the formula for continuous compounding: F = P e r t .
Substitute the given values: P = 1100 , r = 0.058 , and t = 22 into the formula.
Calculate the future value: F = 1100 × e 0.058 × 22 ≈ 3940.508 .
Round the result to the nearest cent: 3940.51 .
Explanation
Understanding the Problem We are given an initial deposit of $1100 in an account with a 5.8% interest rate, compounded continuously. We want to find the balance after 22 years.
Identifying the Formula The formula for continuous compounding is given by: F = P e r t where:
F 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
Substituting the Values We are given:
P = $1100
r = 5.8% = 0.058
t = 22 years Substituting these values into the formula, we get: F = 1100 × e 0.058 × 22 F = 1100 × e 1.276
Calculating the Future Value Calculating the value: F = 1100 × e 1.276 ≈ 1100 × 3.58228 F ≈ 3940.508 Rounding to the nearest cent, we get: F ≈ 3940.51
Final Answer Therefore, the balance after 22 years is approximately $3940.51.
Examples
Continuous compounding is a powerful concept used in finance to model the growth of investments over time. For example, understanding continuous compounding can help you estimate the future value of your retirement savings or the amount you'll owe on a loan. Imagine you deposit money into a high-yield savings account that compounds interest continuously. By using the formula F = P e r t , you can project how much your savings will grow over the years, helping you plan for long-term financial goals such as buying a home or funding your children's education. This calculation provides a more accurate prediction compared to simple or discrete compounding methods.
