To find the total surface area of the composite figure, we need to calculate the surface areas of both the rectangular prism and the triangular prism separately and then combine them while accounting for the areas that are shared (not visible) between the two prisms.
Step 1: Calculate the Surface Area of the Rectangular Prism
The surface area (SA) of a rectangular prism can be calculated using the formula:
\[ SA = 2(lw + lh + wh) \]
Where:
- \( l = 32 , m \) (length)
- \( w = 10 , m \) (width)
- \( h = 8 , m \) (height)
Plugging in the values:
\[ SA = 2(32 \times 10 + 32 \times 8 + 10 \times 8) \]
Calculating each term:
- \( 32 \times 10 = 320 \)
- \( 32 \times 8 = 256 \)
- \( 10 \times 8 = 80 \)
Now sum those products:
\[ 320 + 256 + 80 = 656 \]
Now multiply by 2:
\[ SA = 2 \times 656 = 1312 , m^2 \]
Step 2: Calculate the Surface Area of the Triangular Prism
The surface area of the triangular prism consists of the areas of the two triangular bases and the three rectangular lateral faces.
-
Area of the triangular base:
The base is a right triangle with:
- base (b) = 10 m (equal to the width of the rectangular prism)
- height (h) = 12 m (height of the triangle)
The area of the triangle is:
\[ \text{Area}_{triangle} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 10 \times 12 = 60 , m^2 \]
Since there are two triangular bases:
\[ \text{Total area of triangles} = 2 \times 60 = 120 , m^2 \]
-
Area of the rectangular lateral faces:
The lateral sides consist of three rectangles:
- Two faces correspond to the height of the triangular face (12 m) with bases of 10 m and hypotenuse (20 m).
- The dimensions of these faces are:
- One face with dimensions \( 10 m \times 12 m \)
- One face with dimensions \( 20 m \times 12 m \)
- The third face corresponds to the height of the rectangular prism (8 m) with the length of the rectangular prism (32 m).
Calculating these areas:
\[ \text{Area}_{rectangle1} = 10 \times 12 = 120 , m^2 \]
\[ \text{Area}_{rectangle2} = 20 \times 12 = 240 , m^2 \]
\[ \text{Area}_{rectangle3} = 32 \times 8 = 256 , m^2 \]
Now summing the areas of the rectangles:
\[ 120 + 240 + 256 = 616 , m^2 \]
- Total Surface Area of the Triangular Prism:
Now we sum the area of the triangular bases and the lateral rectangles:
\[ \text{Total SA}_{triangular , prism} = 120 + 616 = 736 , m^2 \]
Step 3: Total Surface Area of the Composite Figure
When combining the surface areas, we must subtract the area of the base of the triangular prism that is in contact with the rectangular prism, which is the area of one triangular base (60 m²).
Thus, the total surface area of the composite figure is:
\[ \text{Total Surface Area} = \text{SA}{rectangular , prism} + \text{SA}{triangular , prism} - \text{Area of base} \]
\[ = 1312 + 736 - 60 = 1988 , m^2 \]
Final Answer:
The total surface area of the composite figure is 1988 m².