A toy is in the form of a cone mounted on a hemisphere. The radius of the hemisphere and that of the cone is 3 cm and height of the cone is 4 cm. The total surface area of the toy (taking `pi = (22)/(7)`) is

Options

A. 75.43 sq cm
B. 103.71 sq. cm.
C. 85.35 sq. cm.
D. 120.71 sq. cm

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 given values
- Radius of the hemisphere (and the cone), \( r = 3 \, \text{cm} \)
- Height of the cone, \( h = 4 \, \text{cm} \)

### Step 2: 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:
\[ l = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \, \text{cm} \]

### Step 3: 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:
\[ \text{CSA}_{\text{cone}} = \frac{22}{7} \times 3 \times 5 = \frac{22 \times 15}{7} = \frac{330}{7} \, \text{cm}^2 \]

### Step 4: Calculate the curved surface area (CSA) of the hemisphere
The formula for the curved surface area of a hemisphere is:
\[ \text{CSA}_{\text{hemisphere}} = 2 \pi r^2 \]
Substituting the values:
\[ \text{CSA}_{\text{hemisphere}} = 2 \times \frac{22}{7} \times 3^2 = 2 \times \frac{22}{7} \times 9 = \frac{396}{7} \, \text{cm}^2 \]

### Step 5: Calculate the total surface area of the toy
The total surface area of the toy is the sum of the curved surface area of the cone and the curved surface area of the hemisphere:
\[ \text{Total Surface Area} = \text{CSA}_{\text{cone}} + \text{CSA}_{\text{hemisphere}} \]
Substituting the values:
\[ \text{Total Surface Area} = \frac{330}{7} + \frac{396}{7} = \frac{726}{7} \, \text{cm}^2 \]

### Step 6: Convert to decimal form
Calculating the decimal value:
\[ \frac{726}{7} = 103.71 \, \text{cm}^2 \]

### Final Answer
The total surface area of the toy is \( 103.71 \, \text{cm}^2 \).

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