Conical caps — sheet required for multiple identical cones: Shantanu’s party cap is a right circular cone with base radius 7 cm and height 24 cm. Find the total sheet area required to make 5 such caps (assume each cap uses only curved surface area and negligible overlap).

Introduction / Context:
Paper or thin-sheet conical caps are made from the lateral (curved) surface of a right circular cone. The needed sheet equals the curved surface area (CSA) of a cone, not the total surface area (which includes the base). For multiple identical caps, multiply the one-cap CSA by the number of caps.

Given Data / Assumptions:

• Radius r = 7 cm.
• Height h = 24 cm.
• Number of caps n = 5.
• Sheet used = lateral surface only (CSA).

Concept / Approach:
Compute slant height l using the Pythagorean relation l = sqrt(r^2 + h^2). Then CSA per cone is π * r * l. Multiply by 5 for five caps. Round sensibly to match the nearest listed option if required.

Step-by-Step Solution:
l = sqrt(7^2 + 24^2) = sqrt(49 + 576) = sqrt(625) = 25 cmCSA (one cap) = π * r * l = π * 7 * 25 = 175π cm2Five caps ⇒ 5 * 175π = 875π ≈ 875 * 3.1416 ≈ 2741.6 cm2Nearest option ≈ 2750 sq cm

Verification / Alternative check:
Using π = 22/7 gives 875 * 22/7 = 2750 cm2 exactly, matching the option.

Why Other Options Are Wrong:
2700 and 3000 are coarse approximations not aligned with π = 22/7; 5000 is far too high for these dimensions.

Common Pitfalls:
Including the base area (πr^2) mistakenly, or using height instead of slant height in the CSA formula.

Final Answer:
2750 sq cm