Question

In the United States, a doll house has the scale of $\mathbf{1}\mathbf{:}\mathbf{12}$of a real house (that is, each length of the doll house is $\frac{\mathbf{1}}{\mathbf{12}}$that of the real house) and a miniature house (a doll house to fit within a doll house) has the scale of $\mathbf{1}\mathbf{:}\mathbf{144}$of a real house. Suppose a real house (Fig. 1-7) has a front length of 20 m, a depth of 12 m, a height of 6.0 m, and a standard sloped roof (vertical triangular faces on the ends) of height 3.0 m. In cubic meters, what are the volumes of the corresponding (a) doll house and (b) miniature house?

Figure 1-7 Problem 32

Short answer

(a). The volume of the doll house is $1.0{\mathrm{m}}^3$.

(b). The volume of the miniature house is $6.0\times {10}^{‐4}{\mathrm{m}}^3$.

Step-by-step solution

Given data

The length of the house,$\mathrm{I}=20\mathrm{m}$

The depth of the house,$\mathrm{d}=12\mathrm{m}$

The height of the house, $\mathrm{h}=6.0\mathrm{m}$

The height of the roof, ${\mathrm{h}}^1=3.0\mathrm{m}$

Understanding the volume

The volume is the space enclosed by a three dimensional object. In this problem, the total volume of the real house is that of a triangular prism in addition to a rectangular box.

The expression for the volume of a rectangular box is given as:

$\mathrm{V}=\mathrm{I}\times \mathrm{d}\times \mathrm{h}$ … (i)

Here,$\mathrm{I}$is the length, d is the breadth and h is the height.

The expression for the volume of triangular prism is given as:

$\mathrm{V}=\frac{1}{2}\left(\mathrm{I}\times \mathrm{d}\times \mathrm{h}\right)$… (ii)

Determination of the volume of real house

Assume that the volume of the sloped roof is ${\mathrm{v}}_{\mathrm{R}}$ and the volume of the box is ${\mathrm{v}}_{\mathrm{B}}$. The total volume $\mathrm{V}$ of the real house is sum of both the volumes.

$$\begin{gathered} \mathrm{V}={\mathrm{v}}_{\mathrm{R}}+{\mathrm{v}}_{\mathrm{B}} \\ =\frac{1}{2}\left(\mathrm{I}\times \mathrm{d}\times {\mathrm{h}}^1\right)+\left(\mathrm{I}\times \mathrm{d}\times \mathrm{h}\right) \end{gathered}$$

Substitute the values in the above equation.

$$\begin{gathered} \mathrm{V}=\left(\frac{1}{2}\times 20\mathrm{m}\times 12\mathrm{m}\times 3.0\mathrm{m}\right) \\ =360{\mathrm{m}}^3+1440{\mathrm{m}}^3 \\ =1800{\mathrm{m}}^3 \end{gathered}$$

Thus, the volume of the real house is $1800{\mathrm{m}}^3$

(a) Calculation of volume a doll house

As the doll house is made with the ratio of 1:12, it means each dimension is reduced by the factor$\frac{1}{12}$. So, the volume of doll house is reduced by ${\left(\frac{1}{12}\right)}^3$.

The volume of the doll house is calculated as:

$$\begin{gathered} {\mathrm{V}}_{\mathrm{Dd}\mathrm{house}}=\mathrm{V}\times {\left(\frac{1}{12}\right)}^3 \\ =1800{\mathrm{m}}^3\times {\left(\frac{1}{12}\right)}^3 \\ =1.0{\mathrm{m}}^3 \end{gathered}$$

Thus, the volume of the doll house is $1.0{\mathrm{m}}^3$.

(b) Determination of the volume of a miniature house

As the miniature house is made with the ratio $1:144$, it means each dimension is reduced by the factor $\frac{1}{144}$. So, the volume is reduced by ${\left(\frac{1}{144}\right)}^3$.

The volume of the miniature house is calculated as:

$$\begin{gathered} {\mathrm{V}}_{\mathrm{miniature}\mathrm{House}}=\mathrm{V}\times {\left(\frac{1}{144}\right)}^3 \\ =1800{\mathrm{m}}^3\times \left(\frac{1}{144}\right) \\ =6.0\times {10}^{‐4}{\mathrm{m}}^3 \end{gathered}$$

Thus, the volume of the miniature hours is $6.0\times {10}^{‐4}{\mathrm{m}}^3$.