https://file.aiquickdraw.com/mathbot/file/1725797784355b6llatkx.jpg

The example in the image provided appears to be related to a **Sinking Fund** problem, specifically Example 35 from a financial mathematics textbook.

Let me break it down:

### Problem Statement:
The question asks to find the amount that needs to be invested every year to accumulate a total of **$90,000** at the end of **10 years**, given that the interest is compounded annually at a rate of **10%**.

### Variables:
- **A**: Future value (amount accumulated) = \$90,000
- **i**: Interest rate per period = 10% = 0.10
- **n**: Number of periods (years) = 10

### Formula Used:
The sinking fund formula used to determine the annual payment $ P $ is:

\[
A = P \times \left( \dfrac{(1 + i)^n - 1}{i} \right)
\]

Where:
- $ A $ is the accumulated amount (future value),
- $ P $ is the periodic payment (what we're solving for),
- $ i $ is the interest rate per period (as a decimal),
- $ n $ is the total number of periods (years).

### Step-by-Step Solution:

1. **Substitute the known values** into the formula:
   \[
   90,000 = P \times \left( \dfrac{(1 + 0.10)^{10} - 1}{0.10} \right)
   \]

2. **Simplify the formula**:
   \[
   90,000 = P \times \left( \dfrac{(1.10)^{10} - 1}{0.10} \right)
   \]

   Using a calculator, $ (1.10)^{10} \approx 2.5937424601 $.

   \[
   90,000 = P \times \left( \dfrac{2.5937424601 - 1}{0.10} \right)
   \]
   \[
   90,000 = P \times \left( \dfrac{1.5937424601}{0.10} \right)
   \]
   \[
   90,000 = P \times 15.937424601
   \]

3. **Solve for P** (the annual payment):
   \[
   P = \dfrac{90,000}{15.937424601}
   \]
   \[
   P \approx 5,647.24
   \]

So, the amount that must be invested each year is approximately **\$5,647.24**.

### Summary:
To accumulate \$90,000 over 10 years at an interest rate of 10% compounded annually, you need to invest **\$5,647.24** each year.

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Do you need further details on any step? Here are 5 questions you might consider exploring:

1. What is the concept of a sinking fund in financial planning?
2. How does compounding affect the amount of annual payments?
3. How would the annual payment change if the interest rate were higher?
4. What other formulas can be used for regular annuities or payments?
5. Can we derive the same result using a different compounding frequency, like monthly compounding?

**Tip:** When dealing with financial calculations, it's useful to familiarize yourself with common financial formulas for future value, present value, and payments.

Learn how to calculate the annual investment required to accumulate $90,000 over 10 years with a 10% annual interest rate, using the sinking fund formula. This example involves compound interest and financial mathematics.

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