It is important to note that
|Mass = Density × Volume; and |
|Volume of model / Volume of miniature = (H of model / H of miniature)3. |
In the above equation, H
is the characteristic dimension (say, height).
If the mass is to be the same, then density is inversely proportional to volume. Also, the volumes are directly proportional to the cubes of the heights for objects that are geometrically similar. Therefore, the heights are seen to be inversely proportional to the cube roots of the densities. Thus,
Height of model = Height of miniature × (Density of miniature / Density of model)1/3
Height of model = 5 × 71/3
= 9.56 inches.
Food for thought:
The above analysis was done for a simple geometry (e.g., pyramid)? Does such an analysis hold for complex shapes?
A lot of scientific and engineering studies are done with scaled-up models (for microscopic phenomena) and scaled-down models (e.g., wind tunnel experiments in aerodynamics). Dimensional Analysis (used above) and Similarity are important concepts in such studies.