Sphere a has a diameter of 12 and is dilated by a scale factor of one half to create sphere b. what is the ratio of the volume of sphere a to sphere b?
2: 1
8: 1
12: 6
144: 36

8:1

Step-by-step explanation:

A diameter of 12 implies a radius of 6.  If this is dilated by a scale factor of 1/2, then the effective radius is 3.

Compare the sphere volumes when r = 6 and r = 3:

VA = (4/3)π(6)^3 and VB = (4/3)π(3)^3 appear in the following ratio:

VA     (4/3)π(216)

=

VB     (4/3)π(27)

and this ratio reduces to 8:1.  This is the ratio of the volume of sphere A to that of sphere B.

The desired ratio is 8:1.

  8 : 1

Step-by-step explanation:

The ratio of volumes is the cube of the ratio of linear dimensions (diameter).

  Sphere A diameter : Sphere B diameter = 2 : 1

  Sphere A volume : Sphere B volume = 2³ : 1² = 8 : 1