Question
\( \$ 1000 \) are deposited in an account with a \( 8.5 \% \) interest rate, compounded continuously. What is the balance after 5 years? \( \mathrm{P}=\$ 1000 \quad r=0.085 \quad \mathrm{t}=5 \) \( \mathrm{~F}= \) Pe \( ^{r t} \quad \mathrm{~F}=\$[?] \)
Ask by Bond Colon. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The balance after 5 years is approximately \$1,529.59.
Solution
To find the balance after 5 years with continuous compounding, we can use the formula for continuous compounding:
\[ F = Pe^{rt} \]
Given:
- Principal amount \( P = \$1000 \)
- Interest rate \( r = 0.085 \) (8.5%)
- Time \( t = 5 \) years
Substitute the given values into the formula:
\[ F = 1000e^{0.085 \times 5} \]
Now, we can calculate the balance after 5 years.
Calculate the value by following steps:
- step0: Calculate:
\(1000\times \exp \left(0.085\times 5\right)\)
- step1: Multiply the numbers:
\(1000\times \exp \left(0.425\right)\)
- step2: Calculate:
\(1000e^{0.425}\)
- step3: Simplify:
\(1000e^{\frac{17}{40}}\)
- step4: Rewrite the expression:
\(1000\sqrt[40]{e^{17}}\)
The balance after 5 years with continuous compounding is approximately $1529.59.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To calculate the balance after 5 years with continuous compounding, we use the formula \( F = Pe^{rt} \). Plugging in the values, we have \( F = 1000 e^{0.085 \times 5} \). Evaluating it gives us \( F \approx 1000 e^{0.425} \approx 1000 \times 1.528 \approx 1528.43 \). Hence, the balance after 5 years will be approximately \( \$1528.43 \). This concept of continuous compounding really takes advantage of the exponential growth of investments! One fascinating aspect is that with continuous compounding, even a small increase in the interest rate can lead to a significantly higher balance over time. So, it’s a great reminder to always look for optimal interest rates when saving or investing.
