To find the volume of the composite figure, we need to calculate the volumes of both the cubes (which form the sides of the house) and the triangular prism (the roof of the house).
-
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, the total volume of the cubes: \[ V_{\text{cubes}} = 2 \times 64 = 128 \text{ in}^3 \]
-
Volume of the triangular prism (roof): The triangular prism has a base that forms a triangle at the top. The base of the triangle is the doorway width (6 inches) and the height of the triangle is 4 inches. The volume \( V \) of a triangular prism is given by: \[ V = \text{Base Area} \times \text{Height} \] The area of the triangle is: \[ \text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 4 = 12 \text{ in}^2 \] The height of the prism is the same as the length of the cubes, which is 4 inches. Therefore, the volume of the triangular prism: \[ V_{\text{prism}} = 12 \text{ in}^2 \times 4 \text{ in} = 48 \text{ in}^3 \]
-
Total Volume: Now, we can add the volume of the cubes and the volume of the triangular prism to find the total volume of the composite figure: \[ V_{\text{total}} = V_{\text{cubes}} + V_{\text{prism}} = 128 \text{ in}^3 + 48 \text{ in}^3 = 176 \text{ in}^3 \]
Thus, the volume of the composite figure is 176 in.³.
Therefore, the correct response is: 176 in.³.