$1250 is deposited in an account with a $6.5\%$ interest rate, compounded continuously. What is the balance after 8 years?
Answer (2)
Use the continuous compounding formula: F = P e r t .
Substitute the given values: P = 1250 , r = 0.065 , and t = 8 .
Calculate the exponent: 0.065 × 8 = 0.52 .
Calculate the future value: F = 1250 × e 0.52 ≈ 2102.53 . The balance after 8 years is $2102.53 .
Explanation
Understanding the Problem We are given an initial deposit of $1250 in an account with a 6.5% interest rate, compounded continuously. We want to find the balance after 8 years.
Identifying the Formula We will use the formula for continuous compounding, which is: F = P e r t where:
F is the future value (the balance after t years),
P is the principal amount (the initial deposit),
r is the annual interest rate (as a decimal),
t is the time in years,
e is the base of the natural logarithm (approximately 2.71828).
Listing Given Values We are given:
P = $1250
r = 6.5% = 0.065
t = 8 years
Substituting Values into the Formula Now, we substitute the given values into the formula: F = 1250 × e 0.065 × 8 First, we calculate the exponent: 0.065 × 8 = 0.52
Calculating the Exponential Term Now we need to calculate e 0.52 . The result of this operation is approximately 1.6820276496988864.
Calculating the Future Value Finally, we multiply this result by the principal amount: F = 1250 × 1.6820276496988864 = 2102.534562123608 Rounding to the nearest cent, we get $2102.53.
Final Answer Therefore, the balance after 8 years is approximately $2102.53.
Examples
Continuous compounding is a concept used in finance to model the growth of an investment when interest is earned constantly and immediately added to the principal. For example, if you invest in a bond that offers continuous compounding, the formula we used helps you project how much your investment will be worth over time. This is also applicable in scenarios like calculating the decay of radioactive materials or modeling population growth, where changes occur continuously.
Using the continuous compounding formula, the balance of $1250 deposited at a 6.5% interest rate after 8 years is approximately $2102.53. This is calculated by substituting the values into the formula, calculating the exponent, and finding the future value. Finally, the result gives us the total amount after the specified time period.
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