Difficulty: Easy
Correct Answer: 550 cm²
Explanation:
Introduction:
This question applies basic solid geometry, specifically the formula for the curved surface area of a right circular cone. You are given the height and slant height, and are asked to compute the lateral (curved) area using a standard formula and a specified value of π.
Given Data / Assumptions:
Concept / Approach:
The curved surface area (CSA) of a right circular cone is given by:
CSA = π * r * lwhere r is the radius of the base, and l is the slant height. Here, we are given h and l. We first find r using the Pythagoras theorem in the right triangle formed by radius, height, and slant height.
Step-by-Step Solution:
Step 1: Use the relation l² = r² + h².Step 2: Substitute values: 25² = r² + 24².Step 3: Compute squares: 625 = r² + 576.Step 4: Rearrange: r² = 625 − 576 = 49.Step 5: Take the square root: r = 7 cm.Step 6: Now apply the CSA formula: CSA = π * r * l = (22/7) * 7 * 25.Step 7: Simplify: (22/7) * 7 = 22, so CSA = 22 * 25 = 550 cm².
Verification / Alternative check:
You can quickly verify by checking the Pythagoras relation again: 7² + 24² = 49 + 576 = 625 = 25². This confirms that r = 7 cm is correct and that the cone dimensions are consistent.
Why Other Options Are Wrong:
Values such as 572 cm², 528 cm², 539 cm², or 600 cm² arise from incorrect radius calculations or from misusing the CSA formula. Only 550 cm² matches the correct evaluation of π * r * l with π = 22/7, r = 7 cm, and l = 25 cm.
Common Pitfalls:
Students sometimes use π * r² (which is the area of a circle) instead of π * r * l for the curved surface area of a cone, or mistakenly use height instead of slant height in the formula. Ensuring you use the correct geometrical formula is crucial for such problems.
Final Answer:
The curved surface area of the cone is 550 cm².
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