To find the total surface area of the composite figure formed by the triangular prism and the rectangular prism, we need to calculate the surface area of each part and then combine them appropriately.
Step 1: Calculate the surface area of the rectangular prism
The formula for the surface area \(SA\) of a rectangular prism is given by: \[ SA = 2lw + 2lh + 2wh \] Where:
- \(l\) is the length,
- \(w\) is the width,
- \(h\) is the height.
For the rectangular prism:
- Length (\(l\)) = 32 m,
- Width (\(w\)) = 10 m,
- Height (\(h\)) = 8 m.
Now plug in these values: \[ SA = 2(32 \times 10) + 2(32 \times 8) + 2(10 \times 8) \] Calculating each term:
- \(2(32 \times 10) = 2(320) = 640\)
- \(2(32 \times 8) = 2(256) = 512\)
- \(2(10 \times 8) = 2(80) = 160\)
Now summing these: \[ SA = 640 + 512 + 160 = 1312 \text{ m}^2 \]
Step 2: Calculate the surface area of the triangular prism
For the triangular prism, the surface area consists of the areas of two triangular faces and three rectangular faces. The total surface area formula for a triangular prism (excluding the bottom face of the triangular prism that sits on the rectangular prism) is: \[ SA = 2 \cdot \text{Area of Triangle} + \text{Perimeter of Triangle} \cdot \text{Height of Prism} \]
Area of the triangle:
The base of the triangle is equal to the width of the rectangular prism, which is 10 meters. The height of the triangular face is given as 12 meters.
The area of the triangular face is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 12 = 60 \text{ m}^2 \]
Total area of the two triangular faces:
\[ 2 \cdot 60 \text{ m}^2 = 120 \text{ m}^2 \]
Perimeter of the triangular face:
The sides of the triangular face are given as 20 meters, and the base is 10 meters. The third side can be calculated using the Pythagorean theorem or directly given as needed: Assuming it's an isosceles triangle, if the two equal sides are 20 m and base 10:
- Side length can be approximated as both \(10^2 + h^2 = 20^2\).
- Height \(h\) is the given 12 m, which fits.
The perimeter: \[ P = 20 + 20 + 10 = 50 \text{ m} \]
Area of Rectangular Faces:
The three rectangular faces are:
- One rectangle of dimensions (base of triangle, height) = \(10 \text{ m} \times 8 \text{ m} = 80 \text{ m}^2\)
- Two rectangles formed from the sides of the triangular prism: Assuming tartan heights leading to each edge considered:
- \(20 \text{ m} \times 8 \text{ m} = 160 \text{ m}^2\)
- A repeat summation applies to its definition leading to: \[ 2 \times (20 \text{ m} \times 8 \text{ m}) = 320 \text{ m}^2 \]
Total area contributions from all:
Putting all the surface area elements together: \[ SA_{triangular} = 120 + 80 + 320 = 520 \text{ m}^2 \]
Final assembly:
Combine both prisms' surface areas: \[ Total SA = SA_{rectangular} + SA_{triangular} - 1 \times Base_{Triangular (hidden)} \] \[ = 1312 + 520 - 60 = 1772 \text{ m}^2 \]
Thus, the total surface area of the figure is: \[ \boxed{1772 \text{ m}^2} \]