To find the surface area of a cone, you need to calculate two areas: the base area and the lateral (or side) area.
The base of the cone is a circle, so to find the area of the base, we use the formula for the area of a circle:
\[ \text{Base Area} = \pi r^2 \]
Where \( r \) is the radius of the circle. Since the diameter is given as 12 meters, the radius is half of that, so:
\[ r = \frac{12}{2} = 6 \, \text{meters} \]
Now we plug that back into our formula for the area of the base:
\[ \text{Base Area} = \pi \times (6 \, \text{meters})^2 = \pi \times 36 \, \text{meters}^2 \]
Next, the lateral area of a cone is given by the formula:
\[ \text{Lateral Area} = \pi r l \]
Where \( l \) is the slant height of the cone, which is given as 15 meters. So we plug in the values for the radius and the slant height:
\[ \text{Lateral Area} = \pi \times 6 \, \text{meters} \times 15 \, \text{meters} = \pi \times 90 \, \text{meters}^2 \]
To find the total surface area of the cone, we add the base area and the lateral area together:
\[ \text{Total Surface Area} = \text{Base Area} + \text{Lateral Area} \]
\[ \text{Total Surface Area} = (\pi \times 36 \, \text{meters}^2) + (\pi \times 90 \, \text{meters}^2) \]
\[ \text{Total Surface Area} = \pi \times (36 + 90) \, \text{meters}^2 \]
\[ \text{Total Surface Area} = \pi \times 126 \, \text{meters}^2 \]
When you evaluate this using the value of \( \pi \), you get approximately 395.84067435231395 square meters.
Thus, the surface area of the cone is about 395.84 square meters. To match this result with the given options, the surface area of the cone is approximately 395.6 square meters, so option 2 is the correct one.
