The curved surface area and the slant height of a right circular cone are 99 square centimetres and 9 cm respectively. What is the diameter of the base of the cone in centimetres?

Introduction / Context:
This question involves a right circular cone for which the curved surface area and the slant height are given. The task is to find the diameter of the base. The problem checks knowledge of the curved surface area formula for a cone and the relationship between radius, slant height, and curved area.

Given Data / Assumptions:

• Curved surface area of the cone A_c = 99 square centimetres.
• Slant height of the cone l = 9 cm.
• Let the base radius be r cm and the diameter be 2r cm.
• We use pi approximately equal to 22 / 7 unless otherwise specified.

Concept / Approach:
The curved surface area of a right circular cone is given by the formula A_c = pi * r * l. We know A_c and l, so we can solve this equation for r. Once r is found, the diameter is simply 2r. This approach uses direct substitution and basic algebra.

Step-by-Step Solution:
Step 1: Write the formula for curved surface area: A_c = pi * r * l.Step 2: Substitute the known values: 99 = pi * r * 9.Step 3: Rearrange to find r: r = 99 / (9 * pi) = 11 / pi.Step 4: Use pi = 22 / 7 to evaluate: r = 11 / (22 / 7) = 11 * 7 / 22 = 77 / 22 = 3.5 cm.Step 5: The diameter of the base is 2r = 2 * 3.5 = 7 cm.

Verification / Alternative check:
Substitute r = 3.5 cm and l = 9 cm back into the curved surface area formula.Compute A_c = pi * r * l = (22 / 7) * 3.5 * 9.Note that 3.5 = 7 / 2, so A_c = (22 / 7) * (7 / 2) * 9 = 22 * 9 / 2 = 198 / 2 = 99 square centimetres.This matches the given curved surface area, confirming that r = 3.5 cm and diameter 7 cm are correct.

Why Other Options Are Wrong:
A diameter of 3.5 cm would correspond to a radius of 1.75 cm, giving a much smaller curved area than 99 square centimetres.Diameters such as 10.5 cm or 14 cm represent larger radii which would give curved surface areas significantly greater than 99 square centimetres for slant height 9 cm.An option like 5.5 cm will not satisfy the exact relation 99 = pi * r * l when evaluated with pi = 22 / 7.

Common Pitfalls:
Learners may confuse the curved surface area formula with the total surface area formula, which includes the base area.Another error is to treat the given 99 as total area and then incorrectly add or subtract base area terms.Misuse of the approximation for pi or algebraic mistakes when solving for r can also lead to incorrect numerical values.

Final Answer:
The diameter of the base of the cone is 7 cm.