b Recall the relationship between the scale factor of two similar solids and the ratio of their areas.
A
a 3:1
B
b 9:1
Practice makes perfect
a The volumes of two similar spheres are 729 in^3 and 27 in^3. We want to first find the ratio of their radii. The ratio of the radii is the same as the ratio of the scale factor. Let's begin by recalling the Areas and Volumes of Similar Solids Theorem.
Areas and Volumes of Similar Solids Theorem
If the scale factor of two similar solids is ab or a:b, then the ratio of their volumes is a^3b^3 or a^3:b^3.
This means that we can form a ratio out of the given volumes of the spheres and the cube root of this ratio will be equal to the scale factor. In this case a^3= 729 is the volume of the larger sphere, and let b^3= 27 be the volume of the smaller sphere.
The ratio of their radii, 3:1, is the same as the scale factor.
b Now that we know that the scale factor for the two spheres we can find the ratio of their surface areas using the Areas and Volumes of Similar Solids Theorem again.
If we let a= 3 and b= 1, the ratio of the squares of a and b will be the ratio of the surface areas of the spheres.
