The curved surface area of a right circular cone is 3080 square centimetres and its slant height is 35 cm. Using the standard mensuration formulas, what is the total surface area of the cone?

Introduction / Context:
This mensuration question deals with a right circular cone. You are given the curved surface area (also called lateral surface area) and the slant height and are asked to find the total surface area. Total surface area of a cone includes both the curved surface area and the area of the circular base. To find it, we first determine the radius of the base from the curved surface area formula, and then add the area of the base to the curved surface area.

Given Data / Assumptions:
- Curved surface area (CSA) of the cone = 3080 square centimetres. - Slant height l of the cone = 35 cm. - For a right circular cone, CSA = π * r * l. - Total surface area (TSA) = π * r * l + π * r^2 = π * r * (l + r). - π is taken as 22 / 7.

Concept / Approach:
We first use the given curved surface area along with the formula CSA = π * r * l to solve for the radius r. Once r is known, we substitute r and the given l into TSA = π * r * (l + r). The arithmetic must be carried out carefully using π = 22 / 7 to get an exact integer result. This approach gives us the total surface area directly in square centimetres.

Step-by-Step Solution:
Step 1: Write the formula for curved surface area of a cone: CSA = π * r * l. Step 2: Substitute CSA = 3080, π = 22 / 7, and l = 35. So 3080 = (22 / 7) * r * 35. Step 3: Simplify the right hand side. First, 35 / 7 = 5, so (22 / 7) * 35 = 22 * 5 = 110. Step 4: The equation becomes 3080 = 110 * r. Step 5: Solve for r by dividing: r = 3080 / 110 = 28 cm. Step 6: Now compute the total surface area using TSA = π * r * (l + r). Step 7: Substitute π = 22 / 7, r = 28, and l = 35. Then TSA = (22 / 7) * 28 * (35 + 28). Step 8: Compute (35 + 28) = 63. So TSA = (22 / 7) * 28 * 63. Step 9: Simplify 28 / 7 = 4. So TSA = 22 * 4 * 63 = 88 * 63. Step 10: Multiply 88 * 63 = 5544 square centimetres.

Verification / Alternative check:
We can quickly verify by computing the base area separately. Base area = π * r^2 = (22 / 7) * 28^2 = (22 / 7) * 784. Simplify 784 / 7 = 112, so base area = 22 * 112 = 2464 sq cm. Adding CSA = 3080 sq cm gives 3080 + 2464 = 5544 sq cm, which matches the TSA obtained earlier. This double check confirms that our value of r and our total surface area are correct.

Why Other Options Are Wrong:
Option 11088 sq cm is exactly double the correct TSA and might come from mistakenly multiplying by 2 at some step. Option 1848 sq cm is much smaller and could arise from misapplying only π * r^2 or mixing up units. Option 2772 sq cm is exactly half of the correct value and may result from forgetting either the curved surface or the base area in the total surface area calculation.

Common Pitfalls:
A common mistake is to confuse curved surface area with total surface area and forget to add the base area. Another error is in the simplification of fractions when using π = 22 / 7, especially not simplifying 35 / 7 or 28 / 7 correctly. To avoid such mistakes, write the formula clearly, substitute values step by step, and verify by recomputing base and curved areas separately.

Final Answer:
The total surface area of the cone is 5544 square centimetres.