To find the total volume of the dollhouse, we need to calculate the volume of both the rectangular prism and the two triangular prisms.
Volume of the Rectangular Prism
The volume \( V \) of a rectangular prism is given by the formula:
\[ V_{\text{rect}} = \text{length} \times \text{width} \times \text{height} \]
From the problem, the dimensions of the rectangular prism are as follows:
- Width = 10 inches
- Height = 12 inches
- The length is not provided directly, but is implied by the bases of the triangular prisms. Each triangular prism has a base length of 8 inches, and since there is a 2-inch gap between the two triangular prisms (total gap is 2 inches), we can assume the length of the rectangular prism is the combined base lengths of the triangular prisms plus the gap.
Thus, the effective length of the rectangular prism can be calculated as follows:
Effective length = base length of triangular prism * number of triangular prisms + gap
= 8 inches + 2 inches = 10 inches
Now we can calculate the volume of the rectangular prism:
\[ V_{\text{rect}} = 10 , \text{in} \times 10 , \text{in} \times 12 , \text{in} = 1200 , \text{in}^3 \]
Volume of One Triangular Prism
The volume \( V \) of a triangular prism is given by the formula:
\[ V_{\text{tri}} = \frac{1}{2} \times \text{base} \times \text{height} \times \text{length} \]
For the triangular prisms, we have:
- Base = 8 inches
- Height (perpendicular height) = 6 inches
- Length = 10 inches (the same as the rectangular prism)
Now we can calculate the volume of one triangular prism:
\[ V_{\text{tri}} = \frac{1}{2} \times 8 , \text{in} \times 6 , \text{in} \times 10 , \text{in} = 240 , \text{in}^3 \]
Total Volume of the Two Triangular Prisms
Since there are two identical triangular prisms, we multiply the volume of one triangular prism by 2:
\[ V_{\text{tri, total}} = 2 \times V_{\text{tri}} = 2 \times 240 , \text{in}^3 = 480 , \text{in}^3 \]
Total Volume of the Dollhouse
Now, we add the volumes of the rectangular prism and the two triangular prisms:
\[ V_{\text{total}} = V_{\text{rect}} + V_{\text{tri, total}} = 1200 , \text{in}^3 + 480 , \text{in}^3 = 1680 , \text{in}^3 \]
Conclusion
The total volume of the dollhouse is:
\[ \boxed{1680 , \text{in}^3} \]
Please check if all parameters are assumed correctly or if any values seem to differ, as the options you provided do not include 1680 in³.