To find the total volume of the dollhouse, we need to calculate the volumes of the rectangular prism and the two triangular prisms separately, and then sum them up.
Step 1: Volume of the rectangular prism
The formula for the volume \( V \) of a rectangular prism is: \[ V = \text{length} \times \text{width} \times \text{height} \] However, we only have the width (10 inches) and height (12 inches). We need to find the length of the rectangular prism.
Since there is a 2-inch gap between the two triangular prisms, and each triangular prism has a base length of 8 inches, we can infer that the total length of the rectangular prism is: \[ \text{length of rectangular prism} = 8 \text{ inches (triangular prism base)} + 2 \text{ inches (gap)} + 8 \text{ inches (triangular prism base)} = 18 \text{ inches} \]
Now we can calculate the volume: \[ V_{\text{rect}} = 18 , \text{in} \times 10 , \text{in} \times 12 , \text{in} = 2160 , \text{in}^3 \]
Step 2: Volume of one triangular prism
The formula for the volume \( V \) of a triangular prism is: \[ V = \frac{1}{2} \times \text{base} \times \text{height} \times \text{length} \] For the triangular prisms:
- Base = 8 inches
- Height = 6 inches
- Length = 10 inches (width of the rectangular prism)
So the volume of one triangular prism is: \[ V_{\text{tri}} = \frac{1}{2} \times 8 , \text{in} \times 6 , \text{in} \times 10 , \text{in} = \frac{1}{2} \times 480 , \text{in}^3 = 240 , \text{in}^3 \]
Since there are two triangular prisms: \[ V_{\text{total_tri}} = 2 \times 240 , \text{in}^3 = 480 , \text{in}^3 \]
Step 3: Total volume of the dollhouse
Now, we add the volumes of the rectangular prism and the two triangular prisms together: \[ V_{\text{total}} = V_{\text{rect}} + V_{\text{total_tri}} = 2160 , \text{in}^3 + 480 , \text{in}^3 = 2640 , \text{in}^3 \]
Therefore, the total volume of the dollhouse is 2,640 cubic inches. The correct response is:
2,640 in.³