Two similar solids have a scale factor of 3 : 5. If the height of solid I is 3 cm, find the height of solid II.

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We are given that two solids, solid I and solid II, are similar with a scale factor of 3:5. The height of solid I is given as 3 cm, and we are asked to find the height of solid II.

A simple definition of similar figures is figures that have the same "shape" but not necessarily the same size. A more rigorous definition is that a similarity is a transformation that preserves angles and maps all lengths in the same ratio (often referred to as the scale factor, the ratio of magnification, or the dilation factor). Another definition is a transformation that preserves ratios of lengths.

If polygons are similar, then corresponding angles are congruent, and corresponding side lengths are in the same ratio, which is the scale factor. "Corresponding lengths" can include the lengths of corresponding sides, diagonals, medians, heights, and so on.

If we assume that the scale factor for a pair of similar figures is a:b, then all corresponding lengths are in the ratio a:b, all corresponding areas are in the ration a² :b² , and all corresponding volumes are in the ratio a³:b³.

For this problem, we have the scale factor as 3:5, so all corresponding lengths, including height, are in a ratio of 3:5. Thus, since the height of solid I is 3 cm, the height of solid II is 5 cm.

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