https://file.aiquickdraw.com/mathbot/file/1727397794551q0xempj4.jpg

This image shows a compound figure made up of a rectangle and a semi-circle. To find the area of the compound figure, we need to calculate the area of each individual shape and then combine them.

### Step 1: Break down the figure
- **Rectangle**: The large rectangle has a height of 9 miles and a total width of \(4 + 3 = 7\) miles.
- **Smaller Rectangle (Cut Out)**: A smaller rectangle is removed from the larger one. This cut-out rectangle has dimensions \(3 \, \text{mi} \times 3 \, \text{mi}\).
- **Semi-circle**: On the right side, there is a semi-circle with a diameter of 8 miles, giving a radius of 4 miles.

### Step 2: Calculate the areas

1. **Area of the large rectangle**:
   \[
   A_{\text{rect}} = \text{width} \times \text{height} = 7 \, \text{mi} \times 9 \, \text{mi} = 63 \, \text{square miles}
   \]

2. **Area of the smaller rectangle**:
   \[
   A_{\text{cut-out}} = \text{width} \times \text{height} = 3 \, \text{mi} \times 3 \, \text{mi} = 9 \, \text{square miles}
   \]
   This area will be subtracted from the total.

3. **Area of the semi-circle**:
   The formula for the area of a circle is \(A = \pi r^2\). For a semi-circle, we divide this by 2:
   \[
   A_{\text{semi-circle}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \times 3.14 \times (4 \, \text{mi})^2 = \frac{1}{2} \times 3.14 \times 16 = 25.12 \, \text{square miles}
   \]

### Step 3: Total area of the compound figure
To get the total area, we add the area of the large rectangle and the semi-circle, then subtract the cut-out area:
\[
\text{Total Area} = A_{\text{rect}} + A_{\text{semi-circle}} - A_{\text{cut-out}}
\]
\[
\text{Total Area} = 63 + 25.12 - 9 = 79.12 \, \text{square miles}
\]

### Final Answer:
The area of the figure is **79.12 square miles**.

Would you like any further details on this? Here are 5 related questions for deeper understanding:

1. How do we calculate the area of a compound figure with more shapes?
2. Why is it necessary to subtract the cut-out area?
3. How would the result change if the semi-circle was a full circle?
4. What is the significance of using \( \pi \approx 3.14 \) in these calculations?
5. Can you think of real-world examples where compound area calculations are useful?

**Tip:** For compound shapes, always carefully identify individual geometric figures before calculating the total area.

This problem involves calculating the area of a compound figure made of a rectangle and a semi-circle. The solution requires using the area formulas for rectangles and semi-circles, and includes subtracting a cut-out rectangle from the total area. This problem is suitable for students in Grades 8-10.

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