To find the total surface area of the composite figure, we need to calculate the surface area of both the rectangular prism and the triangular prism, and then combine them while avoiding double-counting the area where they are joined.
Step 1: Surface Area of the Rectangular Prism
The formula for the surface area of a rectangular prism is given by:
\[ \text{Surface Area} = 2(lw + lh + wh) \]
Where:
- \( l = 32 \) m (length)
- \( w = 10 \) m (width)
- \( h = 8 \) m (height)
Calculating each area:
- \( lw = 32 \times 10 = 320 \) m²
- \( lh = 32 \times 8 = 256 \) m²
- \( wh = 10 \times 8 = 80 \) m²
Now, plug these values into the surface area formula:
\[ \text{Surface Area}_{\text{rectangular}} = 2(320 + 256 + 80) = 2(656) = 1312 \text{ m}^2 \]
Step 2: Surface Area of the Triangular Prism
For the triangular prism, we need to calculate the areas of the two triangular bases and the three rectangular sides.
- Area of the triangular base:
For a triangle, the area is given by:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
The base of the triangular face is the same as the width of the rectangular prism, which is 10 m, and the height is given as 12 m.
\[ \text{Area}_{\text{triangular base}} = \frac{1}{2} \times 10 \times 12 = 60 \text{ m}^2 \]
Since there are two triangular bases, we multiply this area by 2:
\[ \text{Total Area}_{\text{triangular bases}} = 2 \times 60 = 120 \text{ m}^2 \]
- Area of the rectangular sides of the triangular prism:
- The three rectangles correspond to the prism's sides:
- Rectangle 1 (from one side of the triangle to the top edge): Height = 12 m, Width = Length of base = 10 m
- Rectangle 2 (other side): Same as Rectangle 1
- Rectangle 3 (the side opposite to the triangular base): Height needs to follow from the triangular prism’s slant edges, which are indicated as 20 m long.
Calculating each rectangular area:
- Rectangle 1 area = \( 12 \times 10 = 120 \text{ m}^2 \)
- Rectangle 2 area = \( 12 \times 10 = 120 \text{ m}^2 \)
- Rectangle 3 area = \( 20 \times 10 = 200 \text{ m}^2 \)
Total area for the rectangular sides:
\[ \text{Total Area}_{\text{rectangular sides}} = 120 + 120 + 200 = 440 \text{ m}^2 \]
Total Surface Area Calculation for the Triangular Prism:
Total surface area of the triangular prism combines the area of the triangular bases and the rectangular sides:
\[ \text{Surface Area}_{\text{triangular}} = 120 + 440 = 560 \text{ m}^2 \]
Final Calculation
Now we add the surface areas of both prisms, but we should subtract the area of the bottom face of the triangular prism since it is not exposed (it's where it sits on the rectangular prism):
- The area of the supported triangular face is \(60 \text{ m}^2\) (the triangular base area).
Thus, the total exposed surface area of the composite figure is:
\[ \text{Total Surface Area} = \text{Surface Area}{\text{rectangular}} + \text{Surface Area}{\text{triangular}} - \text{Area of contacting face} \]
\[ \text{Total Surface Area} = 1312 + 560 - 60 = 1812 \text{ m}^2 \]
Final Answer
The total surface area of the figure is \( \boxed{1812} \text{ m}^2 \).