Question

Polyhedron 1, which is made up of 810 identical cubes, is similar to Polyhedron 2, which is made up of 30 cubes of that same size. (a) What is the scale factor of Polyhedron 1 to Polyhedron 2?

a) 3:1
b) 27:1
c) 1:3
d) 1:27

Answer 1

The scale factor between Polyhedron 1 and Polyhedron 2 is determined by taking the cube root of the ratio of their cube counts, which results in a scale factor of 27:1. This means that Polyhedron 1 is 27 times larger than Polyhedron 2 in volume, with a linear scale of 3:1.

Step-by-step explanation:

The student asked about the scale factor between Polyhedron 1, which consists of 810 identical cubes, and Polyhedron 2, which is made up of 30 identical cubes. To determine the scale factor, we should consider the ratio of the quantities of cubes that make up each polyhedron. Since the volume of similar solids is proportional to the cube of the scale factor, we need to find the cube root of the ratio of the volumes to determine the linear scale factor.

To find the scale factor between Polyhedron 1 and Polyhedron 2, we calculate the cube root of the ratio of their cube counts:
Scale factor = cube root of (810/30) = cube root of 27 = 3. Therefore, for every 1 cube in Polyhedron 2, there are 3 cubes in Polyhedron 1 in each dimension, which gives a ratio of cubes of 27:1 since 3x3x3 = 27.

The correct answer is (b) 27:1. This means that Polyhedron 1 is 27 times larger than Polyhedron 2 in terms of volume, and the linear scale factor is 3:1.