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An illustration shows a that is shaped like a rectangular prism with two triangular prisms placed on the top, one on each side. The width of the rectangular prism is 10 inches, and the height is 12 inches. The base length of the triangular prisms on the top of the house is 8 inches each. There is a 2-inch gap between the two triangular prisms. The perpendicular height of the triangular prisms is 6 inches. A window is drawn on the face of each triangular prism, and a door is drawn on the bottom center of the rectangular prism.

Write and solve an expression for the total volume of the dollhouse.(1 point)
Responses

2,640 in.3
2,640 in. cubed

264 in.3
264 in. cubed

2,400 in.3
2,400 in. cubed

3,120 in.3
3,120 in. cubed
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Answer 1

To find the total volume of the dollhouse, we need to calculate the volumes of the rectangular prism and the two triangular prisms, then sum them up.

1. **Volume of the rectangular prism:**
The formula for the volume of a rectangular prism is given by:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
Given that the width is \(10\) inches and the height is \(12\) inches, we need to determine the length. The length is not directly given, but we can assume that it’s the distance that allows for the two triangular prisms and the gap between them.

Each triangular prism has a base length of \(8\) inches. Since there is a \(2\)-inch gap between the two triangular prisms, we can calculate the length of the rectangular prism as follows:
\[
\text{length} = 8 + 2 + 8 = 18 \text{ inches}
\]

Now we can find the volume of the rectangular prism:
\[
V_{\text{rectangular}} = 18 \times 10 \times 12 = 2160 \text{ in}^3
\]

2. **Volume of one triangular prism:**
The formula for the volume of a triangular prism is:
\[
V = \frac{1}{2} \times \text{base} \times \text{height} \times \text{length}
\]
Here, the base is \(8\) inches, the height is \(6\) inches, and the length (the length of the triangular prism) is \(10\) inches (since the width of the rectangular prism is also the length of the triangular prisms).

Thus, the volume of one triangular prism is:
\[
V_{\text{triangular}} = \frac{1}{2} \times 8 \times 6 \times 10 = 240 \text{ in}^3
\]

Since there are two triangular prisms, the total volume of the triangular prisms is:
\[
V_{\text{triangular prisms}} = 2 \times 240 = 480 \text{ in}^3
\]

3. **Total volume of the dollhouse:**
Now, we can sum the two volumes:
\[
V_{\text{total}} = V_{\text{rectangular}} + V_{\text{triangular prisms}} = 2160 + 480 = 2640 \text{ in}^3
\]

So the total volume of the dollhouse is:
\[
\boxed{2,640 \text{ in}^3}
\]