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The cross-sections of these solids are sectors. Find the total surface area, rounding to one decimal place.

To find the total surface area of the solid shown in the image, we need to follow these steps:

### 1. **Understand the Shape**:
   - The shape is a prism with a sector (part of a circle) as its cross-section.
   - The base of the prism is a sector with a central angle of 40°.
   - The length of the prism (distance between the two parallel sectors) is 40 cm.
   - The radius of the sector (which also serves as the height of the prism) is 20 cm.

### 2. **Surface Area Calculation**:
   The surface area of this solid includes:
   - The area of the two identical sector bases.
   - The area of the three rectangular faces that wrap around the curved surface of the sector.

### 3. **Area of a Sector**:
   The formula for the area of a sector with radius \( r \) and central angle \( \theta \) (in radians) is:
   \[
   \text{Area of Sector} = \frac{1}{2} r^2 \theta
   \]
   Since \( \theta = 40^\circ = \frac{40\pi}{180} \) radians, the area of each sector is:
   \[
   \text{Area of Sector} = \frac{1}{2} \times 20^2 \times \frac{40\pi}{180} \approx 139.6 \text{ cm}^2
   \]
   So, the area of both sectors combined is:
   \[
   \text{Total Area of Sectors} = 2 \times 139.6 \approx 279.2 \text{ cm}^2
   \]

### 4. **Area of the Rectangular Faces**:
   The three rectangular faces have the following dimensions:
   - **Length 1 (lateral face adjacent to the sector's radius)**: \( 40 \times 20 = 800 \text{ cm}^2 \)
   - **Length 2 (curved surface)**: To find the length of the curved edge of the sector, we use the arc length formula:
     \[
     \text{Arc Length} = r \times \theta = 20 \times \frac{40\pi}{180} \approx 13.96 \text{ cm}
     \]
     So, the area of the curved rectangular face:
     \[
     \text{Area of Curved Face} = 40 \times 13.96 \approx 558.4 \text{ cm}^2
     \]

### 5. **Total Surface Area**:
   Combine the areas calculated:
   \[
   \text{Total Surface Area} = \text{Area of Sectors} + \text{Area of Rectangular Faces} = 279.2 + 800 + 558.4 \approx 1637.6 \text{ cm}^2
   \]

So, the total surface area of the solid is approximately **1637.6 cm²**.

Would you like any more details or have any questions?

Here are five related questions you might be interested in:
1. How do you calculate the volume of this prism?
2. How would the surface area change if the central angle was 60° instead of 40°?
3. Can you explain how to find the length of the curved edge for different radii?
4. What is the significance of the arc length in determining the surface area?
5. How would you find the lateral surface area if the prism was tilted?

**Tip:** When working with angles in geometry, converting between degrees and radians is often necessary for accurate calculations.

Learn how to calculate the total surface area of a prism where cross-sections are sectors. This problem involves geometry concepts such as sector area, prism surface area, and arc length calculation. Suitable for advanced high school students.

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