Two equal cones are touching each other completely at the base circle. Given that the distance between the two vertices is 16 cm and the diameter of the base circle is 12 cm, find the total surface area of this solid.

Answer

To find the total surface area of the solid formed by two equal cones touching at their bases, we can follow these steps:

### Step 1: Identify the given values
- The distance between the vertices of the two cones is 16 cm.
- The diameter of the base circle is 12 cm.

### Step 2: Calculate the radius of the base
The radius \( r \) of the base circle can be calculated from the diameter:
\[ r = \frac{\text{diameter}}{2} = \frac{12 \text{ cm}}{2} = 6 \text{ cm} \]

### Step 3: Determine the height of each cone
Since the total distance between the vertices of the two cones is 16 cm, and both cones are equal, the height \( h \) of each cone is half of this distance:
\[ h = \frac{16 \text{ cm}}{2} = 8 \text{ cm} \]

### Step 4: Calculate the slant height of the cones
The slant height \( l \) of a cone can be calculated using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} \]
Substituting the values of \( h \) and \( r \):
\[ l = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ cm} \]

### Step 5: Calculate the surface area of one cone
The lateral surface area \( A \) of one cone is given by the formula:
\[ A = \pi r l \]
Substituting the values of \( r \) and \( l \):
\[ A = \pi \times 6 \text{ cm} \times 10 \text{ cm} = 60\pi \text{ cm}^2 \]

### Step 6: Calculate the total surface area of both cones
Since there are two cones, the total lateral surface area is:
\[ \text{Total Lateral Surface Area} = 2 \times A = 2 \times 60\pi \text{ cm}^2 = 120\pi \text{ cm}^2 \]

### Step 7: Calculate the area of the base
The area of the base (which is the same for both cones) is:
\[ \text{Area of base} = \pi r^2 = \pi \times 6^2 = 36\pi \text{ cm}^2 \]
Since the bases of the two cones touch each other, we only need to consider the base area once.

### Step 8: Calculate the total surface area of the solid
The total surface area of the solid is the sum of the total lateral surface area and the area of the base:
\[ \text{Total Surface Area} = \text{Total Lateral Surface Area} + \text{Area of base} = 120\pi \text{ cm}^2 + 36\pi \text{ cm}^2 = 156\pi \text{ cm}^2 \]

### Final Answer
The total surface area of the solid is:
\[ \text{Total Surface Area} = 156\pi \text{ cm}^2 \]