To find the total surface area of the composite figure made up of a rectangular prism and a triangular prism, we need to calculate the surface area of each prism separately and then combine their contributions while keeping in mind that we will not count the area where the two prisms are in contact.
Step 1: Surface Area of the Rectangular Prism
The rectangular prism has the following dimensions:
- Length (l) = 32 meters
- Width (w) = 10 meters
- Height (h) = 8 meters
The surface area (SA) of a rectangular prism is given by the formula: \[ SA = 2(lw + lh + wh) \] Calculating each component:
- \(lw = 32 \times 10 = 320\)
- \(lh = 32 \times 8 = 256\)
- \(wh = 10 \times 8 = 80\)
Now, substituting back into the surface area formula: \[ SA = 2(320 + 256 + 80) = 2(656) = 1312 \text{ square meters} \]
Step 2: Surface Area of the Triangular Prism
The triangular prism has a rectangular base aligned with the rectangular prism. The base of the triangular prism has a height of 12 meters, and it is triangular in shape. The sides of the triangular face measure 20 meters.
- Area of the triangular face: The base of the triangle is the same as the width of the rectangular prism (10 meters). The height of the triangle is 12 meters: \[ \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 12 = 60 \text{ square meters} \]
Since the triangular prism has two triangular faces: \[ \text{Total area of triangular faces} = 2 \times 60 = 120 \text{ square meters} \]
- Area of the rectangular sides:
The triangular prism has three rectangular sides. The dimensions for each side are:
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The rectangle formed by the longer side of the triangle and the height of the prism (12 meters tall and 20 meters long): \[ \text{Area} = 20 \times 12 = 240 \text{ square meters} \]
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The two other rectangles correspond to the base of the triangle and the height of the prism (8 meters tall):
- Each of these rectangles has dimensions: (length of the base of the triangle) which is 10 meters and the height of the prism (8 meters): \[ \text{Area of one rectangle} = 10 \times 8 = 80 \text{ square meters} \] Since there are two of these: \[ \text{Total area of two rectangles} = 2 \times 80 = 160 \text{ square meters} \]
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Thus, the total area of the rectangular sides in the triangular prism is: \[ 240 + 160 = 400 \text{ square meters} \]
Step 3: Total Surface Area of the Composite Figure
Now, we can find the total surface area of the composite figure: \[ \text{Total Surface Area} = \text{Surface Area of Rectangular Prism} + \text{Surface Area of Triangular Prism} - \text{Base Area (9 of triangular prism against rectangular prism)} \]
Since the base of the triangular prism (10x8 area = 80 sq m) is not visible: \[ \text{Total Surface Area} = 1312 + 520 - 80 = 1752 \text{ square meters} \]
Thus, the total surface area of the composite figure is: \[ \boxed{1752} \text{ square meters} \]