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 can follow these steps:

### Step 1: Identify the given values
- 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 height of the cone can be calculated by subtracting the radius of the hemisphere from the total height of the toy:
\[ h = H - r = 15.5 \, \text{cm} - 3.5 \, \text{cm} = 12.0 \, \text{cm} \]

### Step 3: Calculate the slant height of the cone
The slant height \( l \) of the cone can be calculated using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} = \sqrt{(3.5)^2 + (12.0)^2} \]
Calculating the squares:
\[ l = \sqrt{12.25 + 144} = \sqrt{156.25} = 12.5 \, \text{cm} \]

### Step 4: Calculate the curved surface area (CSA) of the cone
The formula for the curved surface area of a cone is:
\[ \text{CSA}_{\text{cone}} = \pi r l \]
Substituting the values:
\[ \text{CSA}_{\text{cone}} = \pi \times 3.5 \, \text{cm} \times 12.5 \, \text{cm} \]
Using \( \pi \approx \frac{22}{7} \):
\[ \text{CSA}_{\text{cone}} = \frac{22}{7} \times 3.5 \times 12.5 = \frac{22 \times 3.5 \times 12.5}{7} = \frac{22 \times 43.75}{7} = \frac{968.75}{7} \approx 138.39 \, \text{cm}^2 \]

### Step 5: Calculate the curved surface area of the hemisphere
The formula for the curved surface area of a hemisphere is:
\[ \text{CSA}_{\text{hemisphere}} = 2\pi r^2 \]
Substituting the radius:
\[ \text{CSA}_{\text{hemisphere}} = 2 \pi (3.5)^2 = 2 \pi \times 12.25 \]
Using \( \pi \approx \frac{22}{7} \):
\[ \text{CSA}_{\text{hemisphere}} = 2 \times \frac{22}{7} \times 12.25 = \frac{44 \times 12.25}{7} = \frac{539}{7} \approx 77.00 \, \text{cm}^2 \]

### Step 6: Calculate the total surface area of the toy
The total surface area of the toy is the sum of the curved surface areas of the cone and the hemisphere:
\[ \text{Total Surface Area} = \text{CSA}_{\text{cone}} + \text{CSA}_{\text{hemisphere}} \]
Substituting the values:
\[ \text{Total Surface Area} \approx 138.39 \, \text{cm}^2 + 77.00 \, \text{cm}^2 \approx 215.39 \, \text{cm}^2 \]

### Final Answer
The total surface area of the toy is approximately \( 215.39 \, \text{cm}^2 \).

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