In mensuration, what is the total surface area of a right circular cone with height 14 cm and base radius 7 cm?

Introduction / Context:
This aptitude question tests the formula for the total surface area of a right circular cone, which often appears in competitive exams under the topic of mensuration and geometry. Learners are expected to know how to combine the curved surface area and the base area of a cone, and how to compute the slant height using the Pythagoras theorem. Understanding this concept helps in solving many real life problems that involve conical shapes, such as ice cream cones, funnels, and lampshades.

Given Data / Assumptions:
• Height h of the cone = 14 cm• Radius r of the base = 7 cm• The cone is right circular, so the axis is perpendicular to the base• Use the standard formula for total surface area of a cone

Concept / Approach:
The total surface area of a right circular cone is the sum of its curved surface area and the area of its circular base. The curved surface area is given by π * r * l, where r is the radius and l is the slant height. The base area is π * r^2. The slant height l is found using the Pythagoras theorem in the right triangle formed by the height, radius, and slant height: l^2 = h^2 + r^2.

Step-by-Step Solution:
Step 1: Compute the slant height l.l^2 = h^2 + r^2 = 14^2 + 7^2 = 196 + 49 = 245.l = sqrt(245) cm.Step 2: Write the formula for total surface area (TSA) of the cone.TSA = π * r * l + π * r^2.Step 3: Substitute r = 7 cm and l = sqrt(245).TSA = π * 7 * sqrt(245) + π * 7^2.Step 4: Simplify numerically using π approximately equal to 3.14.TSA ≈ 3.14 * 7 * 15.65 + 3.14 * 49.Step 5: This gives a total surface area close to 498.35 square centimetres.

Verification / Alternative check:
As a quick check, the curved surface area is roughly π * 7 * 15.65 which is a bit more than 340 square centimetres, and the base area π * 49 is about 154 square centimetres. Adding these two approximate values gives around 494 to 500 square centimetres, which confirms that an answer close to 498 square centimetres is reasonable. This supports the selection of 498.35 sq. cm as the most accurate option among those given.

Why Other Options Are Wrong:
Option A: 344.35 sq. cm is too small because it is close only to the curved surface area but ignores the base area contribution.Option B: 462 sq. cm underestimates the total area and does not match the sum of curved and base areas for the given dimensions.Option D: None of these is incorrect because one of the numerical options does match the calculated result quite closely.

Common Pitfalls:
Candidates sometimes forget to include the base area and use only π * r * l, which gives only the curved surface area. Another frequent mistake is to treat the height as the slant height or to apply the Pythagoras theorem incorrectly. Rounding errors can also cause confusion, so it is important to keep enough decimal places while doing intermediate calculations.

Final Answer:
The total surface area of the right circular cone is approximately 498.35 sq. cm.