A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.

Answer

To find the total surface area of the toy, which consists of a cone mounted on a hemisphere, we will follow these steps:

### Step 1: Identify the dimensions
- Radius of the cone and hemisphere, \( r = 3.5 \, \text{cm} \)
- Total height of the toy, \( H = 15.5 \, \text{cm} \)

### Step 2: Calculate the height of the cone
The total height of the toy is the sum of the height of the cone and the radius of the hemisphere. Since the radius of the hemisphere is also the height of the hemisphere, we can express the height of the cone as:
\[ h = H - r = 15.5 \, \text{cm} - 3.5 \, \text{cm} = 12 \, \text{cm} \]

### Step 3: Calculate the slant height of the cone
The slant height \( l \) of the cone can be calculated using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} \]
Substituting the values:
\[ l = \sqrt{12^2 + 3.5^2} = \sqrt{144 + 12.25} = \sqrt{156.25} = 12.5 \, \text{cm} \]

### Step 4: Calculate the surface area of the cone
The lateral surface area of the cone is given by the formula:
\[ \text{Lateral Surface Area of Cone} = \pi r l \]
Substituting the values:
\[ \text{Lateral Surface Area of Cone} = \pi \times 3.5 \times 12.5 \]
Calculating:
\[ \text{Lateral Surface Area of Cone} = \pi \times 43.75 \approx 137.44 \, \text{cm}^2 \]

### Step 5: Calculate the surface area of the hemisphere
The surface area of the hemisphere (excluding the base) is given by the formula:
\[ \text{Surface Area of Hemisphere} = 2 \pi r^2 \]
Substituting the values:
\[ \text{Surface Area of Hemisphere} = 2 \pi (3.5^2) = 2 \pi (12.25) \approx 76.96 \, \text{cm}^2 \]

### Step 6: Calculate the total surface area of the toy
The total surface area of the toy is the sum of the lateral surface area of the cone and the surface area of the hemisphere:
\[ \text{Total Surface Area} = \text{Lateral Surface Area of Cone} + \text{Surface Area of Hemisphere} \]
Substituting the values:
\[ \text{Total Surface Area} \approx 137.44 + 76.96 \approx 214.4 \, \text{cm}^2 \]

### Final Answer:
The total surface area of the toy is approximately \( 214.4 \, \text{cm}^2 \).

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