Question
\( \$ 200 \) are deposited in an account with \( 6 \% \) interest rate, compounded continuously. What is the balance after 5 years? \[ F=\$[?] \] Round to the nearest cent.
Ask by Clark John. in the United States
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The balance after 5 years is \$269.97.
Solution
To calculate the future balance \( F \) with continuous compounding, you can use the formula:
\[
F = P \times e^{rt}
\]
Where:
- \( P \) is the principal amount (\$200)
- \( r \) is the annual interest rate (6% or 0.06)
- \( t \) is the time in years (5 years)
- \( e \) is the base of the natural logarithm (approximately 2.71828)
Plugging in the values:
\[
F = 200 \times e^{0.06 \times 5} = 200 \times e^{0.3} \approx 200 \times 1.34985880758 \approx 269.971761516
\]
Rounding to the nearest cent:
\[
F = \$269.97
\]
\[
F = \$269.97
\]
Reviewed and approved by the UpStudy tutoring team
Like
error msg
Explain 
Simplify this solution Extra Insights
To calculate the future balance for an investment compounded continuously, you can use the formula: \[ F = P e^{rt} \] where: - \( P \) is the principal amount (\$200) - \( r \) is the interest rate (0.06) - \( t \) is the time in years (5) - \( e \) is the base of the natural logarithm, approximately equal to 2.71828. Plugging in the values: \[ F = 200 \times e^{0.06 \times 5} \] Calculating the exponent: \[ 0.06 \times 5 = 0.3 \] Now, calculating \( e^{0.3} \): \[ e^{0.3} \approx 1.34986 \] Now plug this back into the formula: \[ F \approx 200 \times 1.34986 \approx 269.972 \] Rounding to the nearest cent, the balance after 5 years is: \[ F \approx \$269.97 \]
