Question
\( \$ 200 \) are deposited in an account with 60 interest rate, compounded continuously. What is the balance after 5 years? \[ F=\$[?] \]
Ask by Munoz Estrada. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The balance after 5 years is approximately \$4,017.11.
Solution
To calculate the future balance \( F \) of an account 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 (60% or 0.60 as a decimal),
- \( t \) is the time in years (5 years),
- \( e \) is the base of the natural logarithm (approximately 2.71828).
Plugging in the values:
\[
F = 200 \times e^{0.60 \times 5} = 200 \times e^{3}
\]
Calculate \( e^{3} \):
\[
e^{3} \approx 20.0855369232
\]
Now, multiply by the principal:
\[
F = 200 \times 20.0855369232 \approx 4017.11
\]
**Final Balance:**
\[
F = \$4,\!017.11
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To find the balance in an account with continuous compounding, you can use the formula \( F = Pe^{rt} \), where \( P \) is the principal amount, \( r \) is the interest rate, \( t \) is the time in years, and \( e \) is the base of the natural logarithm (approximately equal to 2.71828). In this case, \( P = 200 \), \( r = 0.60 \), and \( t = 5 \). Plugging in these values gives: \[ F = 200 \cdot e^{0.60 \cdot 5} = 200 \cdot e^{3} \approx 200 \cdot 20.0855 \approx 4017.10. \] So, the balance after 5 years is approximately: \[ F = \$4017.10 \]
