Introduction / Context:
Sheet-metal problems for cones typically require the curved surface area (CSA) of a right circular cone, since the base is usually open for containers or funnels. We compute CSA using radius and slant height.
Given Data / Assumptions:
- Height h = 12 cm.
- Base diameter = 14 cm ⇒ radius r = 7 cm.
- Only curved surface area is needed (open cone).
- Use π = 22/7 for exact cancellation with r = 7 cm.
Concept / Approach:
- Slant height l = sqrt(r^2 + h^2).
- Curved surface area: CSA = π*r*l.
Step-by-Step Solution:
r = 7 cm, h = 12 cml = sqrt(7^2 + 12^2) = sqrt(49 + 144) = sqrt(193) ≈ 13.89 cmCSA = π*r*l = (22/7)*7*sqrt(193) = 22*sqrt(193)Since sqrt(193) ≈ 13.89, CSA ≈ 22 * 13.89 ≈ 305.6 cm^2Nearest option ≈ 304 sq cm
Verification / Alternative check:
Using π ≈ 3.14 and l ≈ 13.89: CSA ≈ 3.14*7*13.89 ≈ 305.8 cm^2. Rounding to the nearest provided option gives 304 sq cm, consistent with standard exam rounding.
Why Other Options Are Wrong:
- 504/704/804 sq cm: Significantly larger than precise CSA; these might reflect including a base or using wrong slant height.
- None of these: Incorrect because 304 sq cm matches the computed value by rounding.
Common Pitfalls:
- Confusing height with slant height.
- Including base area when asked for a hollow cone.
- Rounding too early; compute l first then CSA.
Final Answer:
304 sq cm
Discussion & Comments