Final answer:
The ratio of the volumes of the two parts of the cone, when it is divided by a plane through its midpoint, is 1:7. Therefore the correct option is not mention.
Step-by-step explanation:
The question asks for the ratio of the volumes of two parts of a cone that is divided by a plane parallel to its base and through the midpoint of its axis. We will use the concept that the volume of a cone (V) is obtained by the formula V = (1/3)πr²h, where r is the radius of the base and h is the height of the cone.
For the original cone with radius 8 cm and height 12 cm, the volume is:
V = (1/3)π(8 cm)²(12 cm) = 256π cm³
When the cone is cut into two parts, the smaller cone that is formed on top will have half the height of the original cone (because it is cut through the midpoint), which is 6 cm, but the radius is unknown. However, we recognize that similar cones have volumes that scale with the cube of their linear dimensions. Thus, since the height of the smaller cone is half that of the original, its volume is 1/8 of the larger cone.
The ratio of the volume of the smaller cone to the original cone is 1:8, and therefore the ratio of the volume of the larger (bottom) part to the total cone is 7:8. To get the ratio of the volumes of the two parts, we divide the volume of the smaller part by the volume of the bottom part, which is 1:7.
Therefore the correct option is not mention.