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The image shows an investment problem involving compound interest. Here’s what is given:

- **Principal (P)**: $52,250.00
- **Compounded**: Quarterly
- **Time (t)**: 6 years and 3 months (which is 6.25 years)
- **Future Value (A)**: $92,817.56
- **Interest Rate (r)**: Unknown (we are solving for this).

The compound interest formula is:

\[
A = P \left(1 + \frac{r}{n}\right)^{n \cdot t}
\]

Where:
- $ A $ is the amount of money accumulated after $ t $ years, including interest.
- $ P $ is the principal amount (initial deposit or loan balance).
- $ r $ is the annual interest rate (decimal).
- $ n $ is the number of times that interest is compounded per year.
- $ t $ is the number of years.

In this case:
- $ A = 92817.56 $
- $ P = 52250 $
- $ n = 4 $ (since it’s compounded quarterly)
- $ t = 6.25 $

### Step 1: Substitute the values into the formula.

\[
92817.56 = 52250 \left(1 + \frac{r}{4}\right)^{4 \cdot 6.25}
\]

### Step 2: Simplify the equation.

Calculate $ 4 \cdot 6.25 = 25 $:

\[
92817.56 = 52250 \left(1 + \frac{r}{4}\right)^{25}
\]

### Step 3: Solve for $ r $.

1. Divide both sides by 52250:

   \[
   \frac{92817.56}{52250} = \left(1 + \frac{r}{4}\right)^{25}
   \]

2. Take the 25th root of both sides to isolate $ 1 + \frac{r}{4} $.

3. Rearrange to solve for $ r $.

Let me calculate this step for the exact interest rate.The symbolic solution attempt encountered issues due to complexity, so I'll approach this by calculating the interest rate iteratively. 

Let's proceed with a numerical method to approximate the value for $ r $.The interest rate required to grow $52,250.00 to $92,817.56 over 6 years and 3 months, with quarterly compounding, is approximately **9.30%** annually.

Would you like further details on the steps or calculations? 

Here are five related questions you might find useful:
1. How does changing the compounding frequency affect the final amount?
2. What would be the effect of increasing the principal amount on the required interest rate?
3. How would the interest rate change if the investment period were shorter or longer?
4. How does compounding quarterly compare to compounding monthly or annually?
5. How can we use logarithmic functions to solve for the interest rate in compound interest problems?

**Tip:** Compounding more frequently (like monthly or daily) results in a slightly higher amount than annual or quarterly compounding at the same nominal interest rate.

Calculate the interest rate needed to grow an initial investment of $52,250.00 to $92,817.56 over 6 years and 3 months, with interest compounded quarterly.

Learn how to calculate the interest rate needed to grow an investment of $52,250.00 to $92,817.56 over 6 years and 3 months with quarterly compounding. This solution covers the compound interest formula and exponential growth principles, suitable for students in Grades 10-12.

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