To find the volume of the composite figure, we'll break it down into two parts: the volume of the cubes that form the sides of the house and the volume of the triangular prism that forms the roof.
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Volume of the cubes: Each cube has a height, length, and width of \(4\) inches. The volume \(V\) of a cube is given by the formula: \[ V = \text{side}^3 \] For one cube: \[ V = 4^3 = 64 \text{ in}^3 \] Since there are two cubes: \[ \text{Total volume of cubes} = 64 + 64 = 128 \text{ in}^3 \]
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Volume of the triangular prism (roof): The roof of the house is in the shape of a triangular prism. The base of the triangle has a width equal to the width of the cubes (6 inches), and the height of the triangle is 4 inches (as indicated by the dashed line).
The area \(A\) of the triangular base is given by: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] For our triangle: \[ A = \frac{1}{2} \times 6 \times 4 = 12 \text{ in}^2 \]
The height of the prism is the same as the depth of the cubes, which is \(4\) inches. The volume \(V\) of the triangular prism is given by: \[ V = A \times \text{height} \] Therefore: \[ V = 12 \times 4 = 48 \text{ in}^3 \]
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Total volume of the composite figure: Now we can add the volumes together: \[ \text{Total volume} = \text{Volume of cubes} + \text{Volume of roof} \] \[ \text{Total volume} = 128 + 48 = 176 \text{ in}^3 \]
Thus, the volume of the composite figure is 176 in.³.