Question

In the United States, a doll house has the scale of $1:12$ of a real house (that is,each length of the doll house is $\frac{1}{12}$ that of the real house) and a miniature house (a doll house to fit within a doll house) has the scale of $1:44$ of a real house. Suppose a real house has a front length of $20m$, a depth of $12m$, a height of $6.0m$, and a standard sloped roof (vertical triangular faces on the ends) of height $3.0m$ In cubic meters, what are the volumes of the corresponding
miniature house?

Answer 1

From the above figure,

The total volume $V$ of the real house is that of triangular prism (of height $h=3.0m$ and base are $A=20\times 12=240m^2$) in addition to a rectangular box (height $h^{\prime }=6.0m$ and same base). Therefore

$V=\frac{1}{2}hA+h^{\prime }A=(\frac{h}{2}+h^{\prime })A=1800m^3$

From the above figure,

In this case, each dimension (relative to the real house) is reduced by a factor of $1/44$. Therefore,

$V_{miniature}=1800\times \frac{1}{44}=6\times {10}^{-4}{m}^3$