Kimberly invested $ 4 comma 000 in her savings account for 5 years. When she withdrew​ it, she  had $ 4 comma 509.99. Interest was compounded continuously. What was the interest rate on the​ account? Round to the nearest tenth of a percent.


To find the interest rate on the account, we can use the formula for continuous compounding:

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

Where:
- $ A $ is the final amount, $4509.99.
- $ P $ is the initial principal, $4000.
- $ r $ is the interest rate (which we need to find).
- $ t $ is the time, 5 years.
- $ e $ is the mathematical constant approximately equal to 2.71828.

We need to solve for $ r $. Start by substituting the given values into the formula:

\[
4509.99 = 4000 e^{5r}
\]

Now, divide both sides by 4000:

\[
\frac{4509.99}{4000} = e^{5r}
\]

Simplifying:

\[
1.1274975 = e^{5r}
\]

Next, take the natural logarithm of both sides to get rid of the exponent:

\[
\ln(1.1274975) = 5r
\]

Now, solve for $ r $:

\[
r = \frac{\ln(1.1274975)}{5}
\]

Let me calculate that for you.The interest rate on Kimberly's account was approximately 2.4%, rounded to the nearest tenth of a percent.

Would you like more details or have any questions?

Here are 5 related questions:
1. How is continuously compounded interest different from simple interest?
2. What would the final amount be if the interest rate was 3% instead of 2.4%?
3. How can we find the time required for an investment to double under continuous compounding?
4. What is the formula for compound interest with periodic compounding?
5. How would the calculation change if the interest were compounded annually instead of continuously?

**Tip:** Continuous compounding maximizes interest growth since it's calculated at every instant, unlike periodic compounding which occurs at set intervals.

Kimberly invested $4,000 in her savings account for 5 years. When she withdrew it, she had $4,509.99. Interest was compounded continuously. What was the interest rate on the account? Round to the nearest tenth of a percent.

Discover how to calculate the interest rate for Kimberly's $4,000 investment, compounded continuously for 5 years, with a final amount of $4,509.99. This example applies exponential growth, continuous compounding formulas, and natural logarithms. Ideal for high school students in Grades 10-12.

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