$8,900 is invested in an account earning 7% interest (APR), compounded continuously. Write a function showing the value of the account after tt years, where the annual growth rate can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage of growth per year (APY), to the nearest hundredth of a percent.

To solve this problem, we need to develop a function that describes the growth of the investment and calculate the effective annual growth rate (APY).

### Step 1: Write the function for continuous compounding
The formula for the value of an investment with continuous compounding is given by:

\[
A(t) = P \cdot e^{rt}
\]

Where:
- $ A(t) $ is the amount of money accumulated after $ t $ years, including interest.
- $ P $ is the principal amount (initial investment), which is $8,900 in this case.
- $ r $ is the annual interest rate (APR), expressed as a decimal. For 7%, $ r = 0.07 $.
- $ t $ is the time the money is invested for in years.
- $ e $ is the base of the natural logarithm, approximately equal to 2.71828.

So the function becomes:

\[
A(t) = 8900 \cdot e^{0.07t}
\]

### Step 2: Round coefficients to four decimal places
The coefficient $ 0.07 $ is already to four decimal places, so the function remains:

\[
A(t) = 8900 \cdot e^{0.0700t}
\]

### Step 3: Calculate the APY (Annual Percentage Yield)
The APY is the effective annual rate of return, considering the effect of compounding. For continuous compounding, the APY is given by:

\[
\text{APY} = e^r - 1
\]

Substituting $ r = 0.07 $:

\[
\text{APY} = e^{0.07} - 1
\]

Let's calculate this.

\[
\text{APY} \approx 2.71828^{0.07} - 1 \approx 1.072508 - 1 \approx 0.072508
\]

Converting this to a percentage:

\[
\text{APY} \approx 7.25\%
\]

### Final Results

1. **Function describing the investment:**

   \[
   A(t) = 8900 \cdot e^{0.0700t}
   \]

2. **APY (Annual Percentage Yield):**

   \[
   \text{APY} \approx 7.25\%
   \]

This means the investment grows effectively by approximately 7.25% per year, considering continuous compounding.

Would you like more details on any part of the process?

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**Relative Questions:**

1. How would the function change if the interest rate were 5% instead of 7%?
2. What is the value of the investment after 5 years using the function provided?
3. Can you explain the difference between APR and APY in the context of continuous compounding?
4. How does continuous compounding compare to monthly compounding for the same APR?
5. What is the significance of the number $ e $ in the context of continuous compounding?

**Tip:** Continuous compounding maximizes the growth potential of an investment by calculating interest at every possible instant, leading to slightly higher returns compared to other compounding methods.

Learn how to calculate investment growth with continuous compounding for an initial investment of $8,900 earning 7% APR. Includes a step-by-step function derivation and APY calculation. Understand the impact of continuous compounding on investment returns.

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