A solid cone has base radius 9 cm and height 21 cm. It is cut into three parts by two planes parallel to its base at heights 7 cm and 14 cm from the base. What is the ratio of the curved surface areas of the top, middle and bottom parts respectively?

Introduction / Context:
This is a more advanced geometry question involving a right circular cone that is cut into three parts by two planes parallel to the base. We are asked not about volumes but about the curved surface areas of the three resulting parts. Because the planes are parallel to the base, the shapes formed are a smaller cone at the top and two frustums. The key is to use similarity of triangles along the height and to compute the curved surface area of each part using slant heights and radii.

Given Data / Assumptions:
• Original cone radius R = 9 cm. • Original height H = 21 cm. • Cuts are parallel to the base at heights 7 cm and 14 cm from the base. • Therefore, the cone is split into three parts: bottom frustum from 0 to 7 cm, middle frustum from 7 to 14 cm, and top cone from 14 to 21 cm. • We need the ratio of curved surface areas (lateral areas) of top, middle and bottom parts.

Concept / Approach:
Because the cuts are parallel to the base, cross sections at different heights are similar circles, and the triangles formed with the axis are similar as well. The radius at any height from the apex is directly proportional to that height. We can determine the radii at the cuts, then find the corresponding slant heights for each segment. The curved surface area of a cone or frustum is π * r * l or π * (r1 + r2) * s, where r1 and r2 are radii and l or s is the relevant slant height. Since we are looking for a ratio, the common factor π can be ignored.

Step-by-Step Solution:
Step 1: Measure height from the apex instead of from the base. The total height is 21 cm. A cut at 7 cm above the base corresponds to 14 cm from the apex. A cut at 14 cm above the base corresponds to 7 cm from the apex. Step 2: Because radius is proportional to height from the apex, the radius at a height y from the apex is (R / H) * y = (9 / 21) * y = (3 / 7) * y. Step 3: Compute radii: • At y = 7 cm (top cut): r1 = (3 / 7) * 7 = 3 cm. • At y = 14 cm (middle cut): r2 = (3 / 7) * 14 = 6 cm. • At y = 21 cm (base): r3 = (3 / 7) * 21 = 9 cm. Step 4: Now find slant heights from the apex. For the top small cone (0 to 7 cm): h1 = 7, r1 = 3. Slant height l1 = sqrt(r1^2 + h1^2) = sqrt(3^2 + 7^2) = sqrt(9 + 49) = sqrt(58). Step 5: For the cone up to 14 cm: h2 = 14, r2 = 6. Slant height l2 = sqrt(6^2 + 14^2) = sqrt(36 + 196) = sqrt(232). Step 6: For the full cone up to 21 cm: h3 = 21, r3 = 9. Slant height l3 = sqrt(9^2 + 21^2) = sqrt(81 + 441) = sqrt(522). Step 7: Curved surface area of the top small cone = π * r1 * l1. Step 8: Curved surface area of the cone up to 14 cm = π * r2 * l2. Step 9: Curved surface area of the full cone up to 21 cm = π * r3 * l3. Step 10: Middle frustum curved surface area = (cone up to 14 cm) minus (top cone) = π * (r2 * l2 − r1 * l1). Step 11: Bottom frustum curved surface area = (full cone) minus (cone up to 14 cm) = π * (r3 * l3 − r2 * l2). Step 12: For ratio calculations, ignore π and compute r1 * l1, (r2 * l2 − r1 * l1) and (r3 * l3 − r2 * l2). Step 13: Numerical evaluation shows that these three values are in the ratio 1 : 3 : 5 (the algebraic simplification leads to this neat ratio due to similarity).

Verification / Alternative Check:
We can approximate numerically. Compute r1 * l1, r2 * l2 and r3 * l3. The differences produce approximate multiples: the middle frustum area is about three times the top cone area, and the bottom frustum area is about five times the top cone area. This confirms that the ratio of curved surface areas of top, middle and bottom parts is 1 : 3 : 5.

Why Other Options Are Wrong:
• 1 : 4 : 8 and 1 : 6 : 12 suggest simple multiples but do not match the actual geometric relationships derived from similar triangles and slant heights. • 1 : 3 : 9 would imply the bottom part is nine times the top, which contradicts the detailed area calculation.

Common Pitfalls:
One common mistake is to treat heights alone as deciding the curved surface areas, ignoring the change in radius. Another is to apply volume ratios instead of surface area formulas. Also, some students incorrectly use only linear ratios without accounting for the slant height changes. Carefully using similar triangles and the formula for curved surface area of a cone or frustum avoids these issues.

Final Answer:
The ratio of curved surface areas of the top, middle and bottom parts is 1 : 3 : 5.