From a solid wooden cylinder of radius 31.5 cm and height 25 cm, a cone of the…
2026
From a solid wooden cylinder of radius 31.5 cm and height 25 cm, a cone of the same height and same diameter is carved out of it. What is the total surface area (in cm2, rounded off to the nearest tens) of the remaining part of the cylinder?
Answer: D. 12,050 — Concept — When a solid is hollowed out, the boundary of what remains consists of the outer faces that survive plus the newly exposed inner wall of the cavity;…
- A.
15,800
- B.
12,570
- C.
17,630
- D.
12,050
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Show answer & explanation
Correct answer: D
Concept — When a solid is hollowed out, the boundary of what remains consists of the outer faces that survive plus the newly exposed inner wall of the cavity; any face that the cavity completely consumes is no longer part of the boundary. For a right circular cylinder of radius r and height h the curved surface is 2πrh and each flat end is πr2, while for a right circular cone of the same radius and height the curved (lateral) surface is πrl, where the slant height is l = √(r2 + h2).
Applying it to this solid
Here r = 31.5 cm and h = 25 cm. The cone removed has the same radius and the same height, so its circular base coincides exactly with the top face of the cylinder: the whole top disc disappears and a conical pit 25 cm deep is left in its place.
The remaining solid therefore exposes three surfaces: the curved surface of the cylinder, the untouched bottom disc, and the curved surface of the conical cavity.
Slant height of the cavity: l = √(31.52 + 252) = √(992.25 + 625) = √(1617.25) = 40.215 cm (approximately).
Curved surface of the cylinder = 2πrh = 2 × (22/7) × 31.5 × 25 = 4,950 cm2.
Bottom circular face = πr2 = (22/7) × 31.5 × 31.5 = 3,118.5 cm2.
Curved surface of the conical cavity = πrl = (22/7) × 31.5 × 40.215 = 99 × 40.215 = 3,981.3 cm2 (approximately).
Total surface area = 4,950 + 3,118.5 + 3,981.3 = 12,049.8 cm2, which is 12,050 cm2 to the nearest ten.
Surface | Formula | Area (cm2) |
|---|---|---|
Curved surface of the cylinder | 2πrh | 4,950 |
Bottom circular face | πr2 | 3,118.5 |
Curved surface of the conical cavity | πrl | 3,981.3 |
Total | — | 12,049.8 ≈ 12,050 |
Cross-check — Carving out the cone swaps the flat top disc (3,118.5 cm2) for the slanting wall of the pit (3,981.3 cm2), so the answer must exceed the intact cylinder's total surface 2πr(r + h) = 2 × (22/7) × 31.5 × 56.5 = 11,187 cm2 by πr(l − r) = 99 × (40.215 − 31.5) = 863 cm2 approximately, giving 11,187 + 863 = 12,050 cm2. The radius 31.5 = 63/2 is what makes π = 22/7 the intended value here; with π = 3.1416 the same three terms give 12,044.9 cm2, which is nearest to the same offered value.
Result — The remaining solid has a total surface area of about 12,050 cm2.