Violet is going to invest $1,800 and leave it in an account for 6 years. Assuming the interest is compounded continuously, what interest rate, to the nearest tenth of a percent, would be required in order for Violet to end up with $2,400?

To find the interest rate required for Violet to grow her investment from $1,800 to $2,400 in 6 years with continuous compounding, we can use the formula for continuous compound interest:

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

Where:
- $ A $ is the final amount ($2,400),
- $ P $ is the principal or initial investment ($1,800),
- $ r $ is the interest rate (what we need to find),
- $ t $ is the time in years (6 years),
- $ e $ is Euler's number (approximately 2.718).

We can solve for $ r $ as follows:

\[
2400 = 1800 \times e^{6r}
\]

First, divide both sides by 1800:

\[
\frac{2400}{1800} = e^{6r}
\]

\[
\frac{4}{3} = e^{6r}
\]

Now, take the natural logarithm (ln) of both sides:

\[
\ln \left( \frac{4}{3} \right) = 6r
\]

Finally, solve for $ r $:

\[
r = \frac{\ln \left( \frac{4}{3} \right)}{6}
\]

Let's calculate this.The required interest rate for Violet to end up with $2,400 after 6 years, with continuous compounding, is approximately **4.8%**.

Would you like further details on the process or any additional clarification?

Here are five related questions for further exploration:
1. How would the interest rate change if Violet wanted $3,000 instead of $2,400?
2. What would the final amount be if the interest rate were 5% instead of 4.8%?
3. How much interest would Violet earn in total at this rate?
4. What would the required interest rate be if the time was 4 years instead of 6?
5. How does continuous compounding differ from monthly or yearly compounding?

**Tip:** Continuous compounding leads to the highest possible return because it reinvests earnings infinitely frequently.

Discover how to calculate the required interest rate for Violet's $1,800 investment to grow to $2,400 in 6 years using continuous compounding. This problem involves key mathematical concepts like exponential growth and logarithms, and is suitable for high school students in 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