A right circular conical structure stands on a circular base of diameter 21 m and has a vertical height of 14 m. If the cost of colour washing its curved surface is Rs. 6 per square metre (take π = 22/7), what is the total cost of the colour wash?

Introduction / Context:
This problem involves the curved surface area of a right circular cone and a simple cost calculation based on area. It tests your ability to compute slant height from the radius and vertical height, apply the correct surface area formula, and then multiply by the given rate per square metre to get the total cost.

Given Data / Assumptions:

• The conical structure has a circular base with diameter 21 m, so radius r = 21 / 2 = 10.5 m.
• The vertical height of the cone h = 14 m.
• Cost of colour washing the curved surface is Rs. 6 per square metre.
• Use π = 22/7.
• We need the total cost of colour washing the curved surface only.

Concept / Approach:
For a cone, the curved surface area (CSA) is given by CSA = π * r * l, where r is the radius and l is the slant height. The slant height is found from the radius and vertical height using the Pythagorean theorem: l = √(r^2 + h^2). Once we compute the curved surface area, we multiply it by the rate (Rs. 6 per square metre) to get the total cost.

Step-by-Step Solution:
Step 1: Compute the radius: r = 21 / 2 = 10.5 m. Step 2: Use the Pythagorean theorem to find slant height l: l = √(r^2 + h^2) = √(10.5^2 + 14^2). Step 3: Calculate 10.5^2 = 110.25 and 14^2 = 196. So r^2 + h^2 = 110.25 + 196 = 306.25. Step 4: Take square root: l = √306.25 = 17.5 m. Step 5: Compute curved surface area: CSA = π * r * l = (22/7) * 10.5 * 17.5. Step 6: Simplify r * l = 10.5 * 17.5 = (21/2) * (35/2) = 735 / 4 = 183.75. Step 7: Multiply: CSA = (22/7) * 183.75. First 183.75 / 7 = 26.25, then 26.25 * 22 = 577.5 square metres. Step 8: Multiply area by rate: total cost = 577.5 * 6 = 3465 rupees.

Verification / Alternative Check:
You can approximate π as 22/7 and verify the calculations stepwise or use fractional form throughout to avoid decimals. For example, CSA = (22/7) * (735/4) = (22 * 735) / 28 = (22 * 105) / 4 = 2310 / 4 = 577.5. Multiplying by 6 still gives 3465. This confirms that all intermediate steps are consistent and the final cost is correct.

Why Other Options Are Wrong:
Options Rs. 4365, Rs. 4465, and Rs. 3365 correspond to larger or smaller areas than the actual curved surface area when back calculated. They would require incorrect slant heights or misapplied formulas, such as using total surface area instead of curved surface area or miscomputing the radius or height.

Common Pitfalls:
Students sometimes mistakenly use the vertical height instead of the slant height in the curved surface area formula, or they confuse curved surface area with total surface area (which also includes the base). Another common error is using the diameter instead of the radius directly in the CSA formula. Always confirm that you have computed l correctly and that you are using r, not the diameter, inside π * r * l.

Final Answer:
The total cost of colour washing the curved surface is Rs. 3465.