https://r2.erweima.ai/mathbot/image/1723976997665xvb948zt.jpg

Let's solve the surface area problems for the different figures shown in the image.

### **a) L-Shaped Prism**

The L-shaped prism has dimensions:

- Larger section: \(7 \, \text{cm} \times 15 \, \text{cm}\)
- Smaller section: \(7 \, \text{cm} \times 6 \, \text{cm}\)

To find the total surface area, we'll break the shape into rectangular faces and calculate their areas.

1. **Front and back faces:**
   - Front: \(15 \times 7 = 105 \, \text{cm}^2\)
   - Back: Same as the front, \(105 \, \text{cm}^2\)
   
2. **Top and bottom faces:**
   - Top: \(6 \times 7 = 42 \, \text{cm}^2\)
   - Bottom: Larger rectangle minus smaller rectangle (whole bottom):
     - \(15 \times 7 - 6 \times 7 = 105 - 42 = 63 \, \text{cm}^2\)
     
3. **Left and right faces:**
   - Left: \(7 \times 7 = 49 \, \text{cm}^2\)
   - Right: \(7 \times 15 = 105 \, \text{cm}^2\)

Total surface area:
\[
2(105) + 2(42 + 63) + 49 = 210 + 210 + 49 = 469 \, \text{cm}^2
\]

### **b) Rectangular Prism with a Cube on Top**

Dimensions:

- Rectangular prism: \(15 \, \text{cm} \times 8 \, \text{cm} \times 7 \, \text{cm}\)
- Cube: \(3 \, \text{cm} \times 3 \, \text{cm} \times 3 \, \text{cm}\)

1. **Surface area of the rectangular prism:**
   - \(2(lw + lh + wh)\)
   - \(= 2(15 \times 8 + 15 \times 7 + 8 \times 7)\)
   - \(= 2(120 + 105 + 56) = 2(281) = 562 \, \text{cm}^2\)

2. **Surface area of the cube:**
   - \(6 \times 3 \times 3 = 54 \, \text{cm}^2\)
   - **Subtract overlap area:** The bottom face of the cube is not visible, so subtract one face of the cube:
   - \(54 - 9 = 45 \, \text{cm}^2\)

Total surface area:
\[
562 + 45 = 607 \, \text{cm}^2
\]

### **c) Rectangular Prism with a Hollow Section**

Dimensions:

- Outer dimensions: \(10 \, \text{cm} \times 14 \, \text{cm} \times 5 \, \text{cm}\)
- Hollow dimensions: \(7 \, \text{cm} \times 4 \, \text{cm} \times 3 \, \text{cm}\)

1. **Outer surface area (complete prism):**
   - \(2(lw + lh + wh)\)
   - \(= 2(10 \times 14 + 10 \times 5 + 14 \times 5)\)
   - \(= 2(140 + 50 + 70) = 2(260) = 520 \, \text{cm}^2\)

2. **Subtract the hollow part's surface area:**
   - Hollow exposed area: \(2(lw + lh + wh)\)
   - \(= 2(7 \times 4 + 7 \times 3 + 4 \times 3)\)
   - \(= 2(28 + 21 + 12) = 2(61) = 122 \, \text{cm}^2\)

Total surface area:
\[
520 - 122 = 398 \, \text{cm}^2
\]

### **d) Slanted Rectangular Prism**

Dimensions:

- Main dimensions: \(13 \, \text{cm} \times 9 \, \text{cm} \times 7 \, \text{cm}\)
- Slanted section: Reduces by \(5 \, \text{cm} \times 7 \, \text{cm}\)

1. **Main area:**
   - \(2(lw + lh + wh)\)
   - \(= 2(13 \times 9 + 13 \times 7 + 9 \times 7)\)
   - \(= 2(117 + 91 + 63) = 2(271) = 542 \, \text{cm}^2\)

2. **Subtract slanted area:**
   - Missing area: \(5 \times 7 = 35 \, \text{cm}^2\)

Total surface area:
\[
542 - 35 = 507 \, \text{cm}^2
\]

Please check any additional specifics or ask questions.

1. How does subtracting areas work in calculating surface area?
2. What are other shapes that could impact calculations?
3. How do dimensions impact prism surface?
4. What if dimensions change?
5. How do you calculate volume?

**Tip:** Visualize sections for better clarity.

Learn how to calculate the surface area of different geometric shapes including L-shaped prism, rectangular prism with cube, and more. Detailed step-by-step solutions provided.

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