Question
\( \begin{array}{l}\$ 1000 \text { are deposited in an account with a } \\ 8.5 \% \text { interest rate, compounded } \\ \text { continuously. } \\ \text { Now enter } t \text {, or the number of years in the bank. } \\ t=[?]\end{array} \)
Ask by Li Savage. in the United States
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
To find the number of years (\( t \)) it takes for \$1000 to grow to a certain amount with continuous compounding at an 8.5% interest rate, use the formula:
\[
t = \frac{\ln\left(\frac{A}{1000}\right)}{0.085}
\]
Where:
- \( A \) is the desired future amount.
For example, to reach \$2000:
\[
t \approx \frac{\ln(2)}{0.085} \approx 8.15 \text{ years}
\]
Solution
Certainly! Let's break down the problem step by step.
**Given:**
- **Principal (\( P \))**: \$1000
- **Annual Interest Rate (\( r \))**: 8.5% or 0.085 (in decimal)
- **Compounding Method**: Continuously
- **Time (\( t \))**: Number of years (this is what we need to find or work with)
**Continuous Compounding Formula:**
When interest is compounded continuously, the future value (\( A \)) of the investment can be calculated using the formula:
\[
A = P \times e^{rt}
\]
where:
- \( e \) is the base of the natural logarithm (approximately 2.71828)
- \( t \) is the time in years
**Solving for Time (\( t \)):**
If you need to find the number of years (\( t \)) it takes for the investment to reach a certain future value (\( A \)), you can rearrange the formula:
\[
A = 1000 \times e^{0.085t}
\]
To solve for \( t \):
1. **Divide both sides by 1000:**
\[
\frac{A}{1000} = e^{0.085t}
\]
2. **Take the natural logarithm (ln) of both sides:**
\[
\ln\left(\frac{A}{1000}\right) = 0.085t
\]
3. **Solve for \( t \):**
\[
t = \frac{\ln\left(\frac{A}{1000}\right)}{0.085}
\]
**Example:**
Suppose you want to find out how many years it will take for your \$1000 investment to grow to \$2000 with continuous compounding at an 8.5% interest rate.
1. Plug \( A = 2000 \) into the equation:
\[
t = \frac{\ln\left(\frac{2000}{1000}\right)}{0.085} = \frac{\ln(2)}{0.085} \approx \frac{0.6931}{0.085} \approx 8.15 \text{ years}
\]
**Using the Formula to Calculate Future Value:**
If you want to calculate the future value after \( t \) years, simply plug the values into the original formula:
\[
A = 1000 \times e^{0.085t}
\]
**Summary:**
- **To find the future value (\( A \)) after \( t \) years:**
\[
A = 1000 \times e^{0.085t}
\]
- **To find the number of years (\( t \)) it takes to reach a desired amount (\( A \)):**
\[
t = \frac{\ln\left(\frac{A}{1000}\right)}{0.085}
\]
If you have a specific value for \( t \) or \( A \) that you'd like to work with, feel free to provide it, and I can help you with the calculations!
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
Did you know that the formula for continuously compounded interest is given by \( A = Pe^{rt} \)? In this case, \( P \) is the principal amount (\$1000), \( r \) is the interest rate (0.085), and \( A \) is the amount in the account after \( t \) years. By plugging in your values, you can calculate how much money you'll have after any number of years! When working with continuously compounded interest, a common mistake is neglecting to convert the percentage rate into a decimal form. Remember, 8.5% becomes 0.085. Always double-check your calculations, especially when solving for \( t \). It's easy to make arithmetic errors, and the value of money should never be underestimated!
