A toy is in the form of a cone mounted on the hemisphere with the same radius. The diameter of the base of the conical portion is 12 cm and its height is 8 cm. What is the total surface area of the toy ?

Question

Accepted Answer

To determine the total surface area of the toy, which consists of a cone mounted on a hemisphere, we need to calculate the surface area of both the cone and the hemisphere and then add them together.

### Given:
- Diameter of the base of the cone = 12 cm, so the radius $ r $ = 6 cm.
- Height of the cone $ h $ = 8 cm.

### Calculations:

1. **Surface Area of the Cone:**
   - The slant height $ l $ of the cone can be calculated using the Pythagorean theorem:
     $$
     l = \sqrt{r^2 + h^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \ cm
     $$
   - The lateral surface area of the cone is given by:
     $$
     \text{Lateral Surface Area of Cone} = \pi r l = \pi \times 6 \times 10 = 60\pi \ cm^2
     $$

2. **Surface Area of the Hemisphere:**
   - The surface area of a hemisphere is given by:
     $$
     \text{Surface Area of Hemisphere} = 2\pi r^2 = 2\pi \times 6^2 = 72\pi \ cm^2
     $$

3. **Total Surface Area of the Toy:**
   - The total surface area is the sum of the lateral surface area of the cone and the surface area of the hemisphere:
     $$
     \text{Total Surface Area} = 60\pi + 72\pi = 132\pi \ cm^2
     $$

### Conclusion:

- **Option 1: $132 \ \pi \ cm^2$** is correct because the total surface area calculated matches this option.

### Explanation for Incorrect Options:

- **Option 2: $112 \ \pi \ cm^2$** is incorrect because it underestimates the total surface area by not accounting for the correct sum of the cone's lateral surface area and the hemisphere's surface area.
  
- **Option 3: $96 \ \pi \ cm^2$** is incorrect because it significantly underestimates the total surface area, likely due to incorrect calculations or assumptions about the dimensions.

- **Option 4: $66 \ \pi \ cm^2$** is incorrect because it represents only the lateral surface area of the cone, ignoring the hemisphere entirely.

Thus, the correct answer is indeed **Option 1: $132 \ \pi \ cm^2$**.