https://file.aiquickdraw.com/mathbot/file/1728525774552su5tfb08.jpg

The problem in the image can be solved using the formula for continuous compound interest, which is:

\[
A = P e^{rt}
\]

Where:
- $A$ is the final amount ($6000 in this case),
- $P$ is the initial principal ($5000),
- $r$ is the annual interest rate (3.4% or 0.034),
- $t$ is the time in years,
- $e$ is the base of the natural logarithm.

We need to solve for $t$, the time it takes for the investment to grow to $6000.

### Step-by-step Solution:

1. Start with the continuous compounding formula:
   \[
   6000 = 5000 e^{0.034t}
   \]

2. Divide both sides by 5000:
   \[
   \frac{6000}{5000} = e^{0.034t}
   \]
   \[
   1.2 = e^{0.034t}
   \]

3. Take the natural logarithm (ln) of both sides:
   \[
   \ln(1.2) = 0.034t
   \]

4. Solve for $t$:
   \[
   t = \frac{\ln(1.2)}{0.034}
   \]

5. Use a calculator to find the values:
   \[
   \ln(1.2) \approx 0.1823
   \]
   \[
   t \approx \frac{0.1823}{0.034} \approx 5.36
   \]

So, it will take approximately **5.36 years** for the investment to grow from $5000 to $6000.

Would you like further details or have any questions?

Here are some related questions for further practice:
1. What would the time be if the interest rate was 4% instead of 3.4%?
2. What is the final amount if you invest $5000 for 10 years at 3.4% continuously compounded interest?
3. How would the formula change if the interest were compounded annually instead of continuously?
4. What is the time required to double your investment at a rate of 3.4% compounded continuously?
5. How does the continuously compounded interest formula differ from simple interest?

**Tip:** When solving for time in interest problems, always use logarithms to "undo" exponentials.

Suppose an investment earns 3.4% interest compounded continuously. How long will it take an investment of $5000 to be worth $6000? Round your answer to the nearest hundredth.

Calculate the time required for a $5000 investment to grow to $6000 with 3.4% interest compounded continuously. Using exponential growth and logarithmic concepts, this problem helps you understand continuous compound interest. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation