If a right circular cone is divided into two parts by drawing a plane through the midpoint of its axis (parallel to the base), then what is the ratio of the volumes of the two parts (smaller top cone : bottom frustum)?

Introduction / Context:
This problem involves volume relationships in similar solids, specifically in a right circular cone. The cone is cut by a plane parallel to its base and passing through the midpoint of its axis, which creates a smaller cone at the top and a frustum at the bottom. Because the cut is at half the height, the linear dimensions of the smaller cone have a simple proportional relationship with those of the original cone. The question asks for the ratio of the volumes of the two resulting parts.

Given Data / Assumptions:
• Original solid is a right circular cone. • A plane is drawn through the midpoint of the axis, parallel to the base. • This creates two parts: a smaller cone at the top and a frustum at the bottom. • We are asked for the ratio of the volumes of the two parts: top cone : bottom portion.

Concept / Approach:
When a plane cuts a cone parallel to its base, the top part is a smaller cone similar to the original. If the cut is at half the height, then all linear dimensions of the smaller cone (height and radius) are in the ratio 1 : 2 compared to the original cone. For similar solids, the ratio of their volumes is the cube of the ratio of their corresponding linear dimensions. Therefore, the volume ratio of the smaller cone to the original cone is (1/2)^3 = 1/8. The bottom part is the remaining volume, which is the original volume minus the top cone volume.

Step-by-Step Solution:
Step 1: Let the height of the original cone be H and the radius of its base be R. Step 2: The plane passes through the midpoint of the axis, so the height of the smaller top cone is H/2 and its base radius is R/2, due to similarity. Step 3: Volume of a cone is (1/3) * π * r^2 * h. So the volume of the original cone is V = (1/3) * π * R^2 * H. Step 4: Volume of the smaller cone is V_small = (1/3) * π * (R/2)^2 * (H/2) = (1/3) * π * (R^2 / 4) * (H / 2) = (1/3) * π * R^2 * H / 8 = V / 8. Step 5: So the smaller cone has one eighth of the volume of the original cone. Step 6: Volume of the bottom frustum is V_frustum = V − V_small = V − V/8 = (7/8) V. Step 7: Therefore, the ratio of the volumes of the two parts (top cone : bottom frustum) is (V/8) : (7V/8) = 1 : 7.

Verification / Alternative Check:
We can plug in specific values to verify. Let R = 2 units and H = 2 units. Then the original volume is V = (1/3) * π * 4 * 2 = (8/3) π. The cut at half height gives a smaller cone with radius 1 and height 1, whose volume is (1/3) * π * 1 * 1 = (1/3) π. The difference is bottom volume = (8/3) π − (1/3) π = (7/3) π. The ratio top : bottom is (1/3) π : (7/3) π = 1 : 7, confirming our result.

Why Other Options Are Wrong:
• 1 : 2 and 1 : 4 may arise from mistakenly using linear or area ratios instead of volume ratios. • 1 : 8 is the ratio of the top cone volume to the total cone volume, not to the remaining frustum volume.

Common Pitfalls:
A frequent error is confusing the relationship between linear dimensions and volume in similar solids. Some learners use the same ratio for volume as for height or incorrectly square instead of cubing the linear ratio. Always remember that for similar three-dimensional figures, volumes scale with the cube of the linear scale factor. Another pitfall is interpreting the question as asking for top cone : whole cone instead of top cone : bottom part.

Final Answer:
The ratio of the volumes of the two parts (top cone to bottom frustum) is 1 : 7.