A toy is in the. shape of a hemisphere surmounted by a cone. If radius of base.of the cone is 3 cm and height is 4cm. The total surface area of the toy is:

Options

A. `33 pi cm^2`
B. `42 pi cm^2`
C. `66 pi cm^2`
D. `56 pi cm^2`

Explanation

To find the total surface area of the toy, which is shaped like a hemisphere surmounted by a cone, we can follow these steps:

### Step 1: Identify the given values
- Radius of the base of the cone (r) = 3 cm
- Height of the cone (h) = 4 cm

### Step 2: Calculate the slant height of the cone
The slant height (l) of the cone can be calculated using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
Substituting the values:
\[ l = \sqrt{3^2 + 4^2} \]
\[ l = \sqrt{9 + 16} \]
\[ l = \sqrt{25} \]
\[ l = 5 \text{ cm} \]

### Step 3: Calculate the surface area of the cone
The formula for the curved surface area of a cone is given by:
\[ \text{Curved Surface Area of Cone} = \pi r l \]
Substituting the values:
\[ \text{Curved Surface Area of Cone} = \pi \times 3 \times 5 \]
\[ \text{Curved Surface Area of Cone} = 15\pi \text{ cm}^2 \]

### Step 4: Calculate the surface area of the hemisphere
The formula for the curved surface area of a hemisphere is:
\[ \text{Curved Surface Area of Hemisphere} = 2\pi r^2 \]
Substituting the value of r:
\[ \text{Curved Surface Area of Hemisphere} = 2\pi \times 3^2 \]
\[ \text{Curved Surface Area of Hemisphere} = 2\pi \times 9 \]
\[ \text{Curved Surface Area of Hemisphere} = 18\pi \text{ cm}^2 \]

### 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:
\[ \text{Total Surface Area} = \text{Curved Surface Area of Cone} + \text{Curved Surface Area of Hemisphere} \]
Substituting the values:
\[ \text{Total Surface Area} = 15\pi + 18\pi \]
\[ \text{Total Surface Area} = 33\pi \text{ cm}^2 \]

### Final Answer
The total surface area of the toy is \( 33\pi \text{ cm}^2 \).

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