Proportional scaling is used to change the size of an object without changing its shape. For example, a model train has the same shape as a real train; as a scaled-down version, it is just much smaller. Scale drawings and models are commonly used in applications such as architecture and engineering, when it is useful to have a smaller representation of a larger object.

A proportion represents two equal ratios, written as a/b = c/d or a:b = c:d. When you proportionately scale an object, you multiply each linear dimension by the same number (this number is called a scale factor). For example, if you wanted to enlarge a photograph, you would multiply both the length and width by the same number (greater than one). The new image would be bigger but would have the same shape as the original: the ratio of length to width for both images would be equal. If the dimensions were not changed by the same factor, the new image would be a different shape and would not be proportionately scaled. For an example, take a look at this Proportional Scaling Illustration.

What happens when you enlarge a cube? If the original cube has a side length, a, of 1 inch, it has a volume of 1 cubic inch (volume = length • width • height = a³). Each surface has an area of 1 square inch (area = length • width = a²), and since there are six faces, the cube's total surface area is 6 square inches (total surface area = 6a²). If you multiply each linear dimension—the length, width, and height—of the original cube by a factor of 2, you'll end up with a proportionately scaled larger cube. However, if you compare the original cube to the new cube, you'll see that the ratios comparing side length, surface area, and volume are not proportional.

In other words, as an object is scaled up (as its size increases), its volume grows faster than its surface area. So, side length, surface area, and volume do not scale proportionately. This relationship has implications in many areas, such as engineering and biomechanics. For example, the stress on an object increases with size; this explains why skyscrapers cannot be made out of wood like smaller buildings can, and why sand castles can only be constructed to a limited height. As seen in the video, heat generation is related to volume, while heat loss is related to surface area. This explains why a small jet engine cannot simply be a smaller version of a big jet engine (it would lose too much heat), and why large animals do better than small animals in cold climates (large animals lose heat less quickly).

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