Question
\( \$ 900 \) are deposited in an account with \( 4 \% \) interest rate, compounded continuously. What is the balance after 10 years? \[ F=\$[?] \]
Ask by Horton Gibbs. in the United States
Jan 20,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The balance after 10 years is approximately \$1,342.64.
Solution
To find the balance after 10 years with continuous compounding, we can use the formula for continuous compounding:
\[ F = P \times e^{rt} \]
Where:
- \( F \) is the final balance
- \( P \) is the principal amount (initial deposit)
- \( r \) is the interest rate
- \( t \) is the time in years
Given:
- Principal amount \( P = \$900 \)
- Interest rate \( r = 4\% = 0.04 \)
- Time \( t = 10 \) years
Substitute the values into the formula:
\[ F = 900 \times e^{0.04 \times 10} \]
Now, we can calculate the final balance.
Calculate the value by following steps:
- step0: Calculate:
\(900e^{0.04\times 10}\)
- step1: Multiply the numbers:
\(900e^{0.4}\)
- step2: Convert the expressions:
\(900e^{\frac{2}{5}}\)
- step3: Rewrite the expression:
\(900\sqrt[5]{e^{2}}\)
The balance after 10 years with continuous compounding is approximately $1342.64.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To find the balance after 10 years with continuous compounding, you can use the formula \( F = Pe^{rt} \), where \( P \) is the principal amount, \( r \) is the interest rate, and \( t \) is the time in years. Here, \( P = 900 \), \( r = 0.04 \), and \( t = 10 \). Plugging in the values, we get: \[ F = 900 \cdot e^{0.04 \cdot 10} \] Calculating \( e^{0.4} \approx 1.4918 \): \[ F \approx 900 \cdot 1.4918 \approx 1342.62 \] Thus, the balance after 10 years is approximately \( \$1342.62 \).
