Question II. Meghan invested a sum of $5,000 in an account ten years ago. She earned a changing (over time) interest rate on her investment. She earned 4% annually compounded over the first 3 years, 4% quarterly compounded over the next 2 years, and 3% percent continuously compounded thereafter. What is her investment worth today? Show all work.

We are given that Meghan invested $5,000 and earned interest at varying rates over different periods of time. The interest rates are applied with different compounding methods over three distinct periods:

1. **4% annually compounded over the first 3 years.**
2. **4% quarterly compounded over the next 2 years.**
3. **3% continuously compounded for the remaining 5 years.**

We need to calculate how much her investment is worth today. We'll break it down into three stages, calculate the investment value at the end of each stage, and then move on to the next stage.

### Step 1: First 3 Years – Annual Compounding

The formula for compound interest is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- $A$ is the amount after interest,
- $P$ is the principal (initial investment),
- $r$ is the annual interest rate (in decimal form),
- $n$ is the number of compounding periods per year,
- $t$ is the time in years.

For the first 3 years:
- $P = 5000$,
- $r = 0.04$ (4%),
- $n = 1$ (compounded annually),
- $t = 3$.

Substituting these values into the formula:
\[
A_1 = 5000 \left(1 + \frac{0.04}{1}\right)^{1 \times 3} = 5000 (1.04)^3
\]

### Step 2: Next 2 Years – Quarterly Compounding

For the next 2 years, the interest is compounded quarterly. The same formula applies, but with $n = 4$ for quarterly compounding and $t = 2$ years.

We use the amount $A_1$ from the first stage as the new principal:
- $P = A_1$,
- $r = 0.04$,
- $n = 4$,
- $t = 2$.

The formula becomes:
\[
A_2 = A_1 \left(1 + \frac{0.04}{4}\right)^{4 \times 2} = A_1 \left(1 + 0.01\right)^8 = A_1 (1.01)^8
\]

### Step 3: Last 5 Years – Continuous Compounding

For the last 5 years, interest is compounded continuously. The formula for continuous compounding is:
\[
A = P e^{rt}
\]
Where:
- $e$ is Euler's number (approximately 2.718),
- $r$ is the annual interest rate (in decimal form),
- $t$ is the time in years.

We use the amount $A_2$ from the second stage as the new principal:
- $P = A_2$,
- $r = 0.03$ (3%),
- $t = 5$.

The formula becomes:
\[
A_3 = A_2 e^{0.03 \times 5} = A_2 e^{0.15}
\]

### Calculation

Let's now calculate the values step by step. I will compute each stage.Here are the results after each stage of compounding:

1. **After the first 3 years (4% annual compounding):**  
   Meghan's investment grows to $ A_1 = 5624.32 $.

2. **After the next 2 years (4% quarterly compounding):**  
   Her investment increases to $ A_2 = 6090.33 $.

3. **After the final 5 years (3% continuous compounding):**  
   Her investment reaches $ A_3 = 7075.96 $.

Therefore, Meghan's investment is worth **$7,075.96** today.

Would you like further details or explanations?

Here are 5 related questions you might explore:
1. How would the final value change if the first 3 years had 5% annual compounding instead?
2. What if the quarterly compounding rate was 3.5% instead of 4%?
3. How does continuous compounding compare to annual compounding over a longer period?
4. What would be the total investment after 10 years if the entire period used continuous compounding at 4%?
5. How would monthly compounding affect the investment if applied during the second stage?

**Tip:** Continuous compounding grows faster than other methods as it assumes interest is being applied constantly, leading to exponential growth.

Meghan invested a sum of $5,000 in an account ten years ago. She earned a changing (over time) interest rate on her investment. She earned 4% annually compounded over the first 3 years, 4% quarterly compounded over the next 2 years, and 3% percent continuously compounded thereafter. What is her investment worth today? Show all work.

Discover how to calculate compound interest for Meghan's $5,000 investment with varying interest rates over 10 years. The investment involves 4% annually compounded for 3 years, 4% quarterly compounded for 2 years, and 3% continuously compounded for the remaining 5 years. Learn step-by-step calculations for each stage and understand how compound and continuous interest affect the final value. Suitable for Grades 10-12.

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