To calculate the total surface area of the composite figure, we will find the surface areas of both the rectangular prism and the triangular prism separately, and then combine them while accounting for the shared base area.
Step 1: Surface Area of the Rectangular Prism
The formula for the surface area \( A \) of a rectangular prism is given by: \[ A = 2(lw + lh + wh) \] where \( l \) is the length, \( w \) is the width, and \( h \) is the height.
Given:
- Length (\( l \)) = 32 meters
- Width (\( w \)) = 10 meters
- Height (\( h \)) = 8 meters
Calculating the surface area: \[ A_{\text{rectangular}} = 2(32 \times 10 + 32 \times 8 + 10 \times 8) \] Calculating each term: \[ 32 \times 10 = 320 \] \[ 32 \times 8 = 256 \] \[ 10 \times 8 = 80 \] Now, summing these: \[ 320 + 256 + 80 = 656 \] Finally, \[ A_{\text{rectangular}} = 2 \times 656 = 1312 \text{ m}^2 \]
Step 2: Surface Area of the Triangular Prism
The formula for the surface area of a triangular prism is given by: \[ A = \text{Area of the two triangular bases} + \text{Area of the three rectangular sides} \]
Area of the triangular bases: The triangular prism has two triangular faces. The area \( A \) of each triangle can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Given:
- The base of the triangular face is equal to the width of the rectangular prism = 10 meters.
- The height of the triangular face = 12 meters.
Calculating the area of one triangle: \[ \text{Area of one triangle} = \frac{1}{2} \times 10 \times 12 = 60 \text{ m}^2 \] Thus, the area of two triangular bases: \[ \text{Area of two triangles} = 2 \times 60 = 120 \text{ m}^2 \]
Area of the rectangular sides: The prism has three rectangular sides:
- The base side (the same as the width of the rectangular prism) = width × height = \( 10 \times 12 = 120 , \text{m}^2 \).
- The two sides where the height of the prism comes into play. The triangular faces are aligned at \( 20 , \text{meters} \) each.
Calculating the areas of these sides:
- Side 1 (the height aligned): \( 12 \times 20 = 240 , \text{m}^2 \)
- Side 2 (the back/other height aligned): \( 12 \times 20 = 240 , \text{m}^2 \)
Total rectangular sides area: \[ \text{Area of sides} = 120 + 240 + 240 = 600 \text{ m}^2 \]
Total Surface Area of the Triangular Prism
Now combining both parts, we find the total surface area of the triangular prism: \[ A_{\text{triangular}} = 120 + 600 = 720 \text{ m}^2 \]
Step 3: Total Surface Area of the Composite Figure
When adding both prisms together, we need to subtract the area of the base of the triangular prism that coincides with the top face of the rectangular prism: Area of the base triangle equals the area of one triangle base calculated earlier = \( 60 , \text{m}^2 \).
Thus, the total surface area \( A_{\text{total}} \): \[ A_{\text{total}} = A_{\text{rectangular}} + A_{\text{triangular}} - \text{Area of one triangular base} \] \[ A_{\text{total}} = 1312 + 720 - 60 = 1972 \text{ m}^2 \]
Therefore, the total surface area of the composite figure is: \[ \boxed{1972} , \text{m}^2 \]