To find the total surface area of the composite figure (a triangular prism on top of a rectangular prism), we can follow these steps:
1. Calculate the surface area of the rectangular prism.
The surface area \( S \) of a rectangular prism can be calculated using the formula: \[ S = 2lw + 2lh + 2wh \] where:
- \( l \) is the length,
- \( w \) is the width,
- \( h \) is the height.
For the rectangular prism:
- Length \( l = 32 , \text{m} \)
- Width \( w = 10 , \text{m} \)
- Height \( h = 8 , \text{m} \)
Plugging in the values: \[ S_{\text{rectangular}} = 2(32)(10) + 2(32)(8) + 2(10)(8) \] Calculating each term: \[ = 2(320) + 2(256) + 2(80) \] \[ = 640 + 512 + 160 \] \[ = 1312 , \text{m}^2 \]
2. Calculate the surface area of the triangular prism.
The surface area of a triangular prism can be calculated using the formula: \[ S = bh + P_b h \] where:
- \( b \) is the area of the triangular base,
- \( P_b \) is the perimeter of the base,
- \( h \) is the height of the prism (length of the prism).
Step 2a: Find the area of the triangular base. The area \( A \) of a triangle can be calculated as: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Assuming the triangular base has:
- Base \( = 10 , \text{m} \) (equal to the width of the rectangle)
- Height \( = 12 , \text{m} \)
\[ A = \frac{1}{2} \times 10 \times 12 = 60 , \text{m}^2 \]
Step 2b: Find the perimeter of the triangular base. The sides of the triangle measure:
- Two sides \( = 20 , \text{m} \) (equal upper edges)
- One base \( = 10 , \text{m} \)
The perimeter \( P_b \) is: \[ P_b = 20 + 20 + 10 = 50 , \text{m} \]
Step 2c: Calculate the surface area of the triangular prism. The height of the triangular prism is the same as the length of the rectangular prism \( = 32 , \text{m} \).
Now calculating: \[ S_{\text{triangular}} = A + P_b h \] \[ = 60 + (50)(32) \] \[ = 60 + 1600 = 1660 , \text{m}^2 \]
3. Combine the surface areas.
Now combine the surface areas of both prisms. However, since the top triangular surface and the bottom rectangular surface (the area of the rectangle that is covered by the triangular prism) are touching, we must subtract the area of the base of the triangular prism from the total:
The area of the base (bottom triangle where the top sits):
- \( = 60 , \text{m}^2 \) (the area of the triangle on the top)
Thus, the total surface area \( S_{total} \): \[ S_{total} = S_{\text{rectangular}} + S_{\text{triangular}} - \text{Area of the triangle on the bottom} \] \[ = 1312 + 1660 - 60 \] \[ = 2912 , \text{m}^2 \]
Conclusion
The total surface area of the composite figure is: \[ \boxed{2912 , \text{m}^2} \]