Cone Surface Area Calculation

This solution explains how to find the total surface area of a cone given its radius and slant height.

Identify Given Values

• The radius of the cone is given as $r_{cone} = 2r$.
• The slant height of the cone is given as $l = \frac{1}{2}$.

Cone Surface Area Formula

The formula for the total surface area (TSA) of a cone is:

$ \text{TSA} = \pi \times \text{radius} \times (\text{radius} + \text{slant height}) $

Using the variables provided:

$ \text{TSA} = \pi \times r_{cone} \times (r_{cone} + l) $

Perform Calculation

Substitute the given values into the formula:

1. Substitute $r_{cone} = 2r$ and $l = \frac{1}{2}$: $ \text{TSA} = \pi \times (2r) \times (2r + \frac{1}{2}) $
2. Simplify the expression inside the parentheses: $2r + \frac{1}{2} = \frac{4r}{2} + \frac{1}{2} = \frac{4r + 1}{2}$.
3. Substitute this back into the TSA equation: $ \text{TSA} = \pi \times (2r) \times \left(\frac{4r + 1}{2}\right) $
4. Cancel out the 2 in the numerator and denominator: $ \text{TSA} = \pi \times r \times (4r + 1) $
5. The final expression for the total surface area is: $ \text{TSA} = \pi r (4r + 1) $

Final Answer Derivation

The calculated total surface area is $\pi r (4r + 1)$. This matches Option C.