The curved surface area of a right circular cone is 10010 square centimetres and its slant height is 91 centimetres. Find the total surface area of the cone in square centimetres.

Introduction / Context:
This mensuration question deals with a right circular cone and distinguishes between curved surface area and total surface area. Total surface area includes both the curved surface and the circular base, and is commonly required when calculating materials used to cover a cone shaped object.

Given Data / Assumptions:

• Curved surface area (CSA) of the cone = 10010 square centimetres.
• Slant height l = 91 cm.
• We must find the total surface area (TSA) in square centimetres.
• Use pi = 22 / 7.

Concept / Approach:
For a right circular cone:
CSA = pi * r * l.
Base area = pi * r^2.
Total surface area TSA = CSA + base area.
We first use CSA and l to find the radius r, then compute the base area and add it to the curved surface area.

Step-by-Step Solution:
Step 1: Use CSA formula: CSA = pi * r * l. Step 2: Substitute CSA = 10010, l = 91, and pi = 22 / 7. Step 3: So 10010 = (22 / 7) * r * 91. Step 4: Simplify (22 / 7) * 91 = 22 * 13 = 286. Step 5: Hence 10010 = 286 * r. Step 6: Solve for r: r = 10010 / 286 = 35 cm. Step 7: Now find base area: base area = pi * r^2 = (22 / 7) * 35^2. Step 8: 35^2 = 1225, so base area = (22 / 7) * 1225 = 22 * 175 = 3850 square centimetres. Step 9: Total surface area TSA = CSA + base area = 10010 + 3850 = 13860 square centimetres.

Verification / Alternative check:
We can recheck the curved surface area using r = 35 cm and l = 91 cm. CSA = pi * r * l = (22 / 7) * 35 * 91. Simplifying 35 / 7 = 5, we get CSA = 22 * 5 * 91 = 110 * 91 = 10010 square centimetres, which matches the given value. Therefore our value of r and hence TSA are consistent.

Why Other Options Are Wrong:
6930 and 4620 square centimetres: These are significantly smaller than the correct TSA and could arise from using only CSA or only base area incorrectly.
27720 square centimetres: This is exactly double 13860 and might come from mistakenly adding CSA twice.
10010 square centimetres: This is only the curved surface area without the base, not the total surface area.

Common Pitfalls:
Students often confuse curved surface area with total surface area and forget to add the base area. Others may misuse the formula for CSA or incorrectly simplify the fraction involving pi. Always clearly distinguish between CSA and TSA and carefully work through the algebra when solving for the radius.

Final Answer:
The total surface area of the cone is 13860 square centimetres.