A toy is in the shape of a right circular cylinder with a hemisphere on one end and a cone on the other. The radius and height of the cylindrical part are 5 cm and 12 cm respectively. The radii of the hemispherical and conical parts are the same as that of the cylindrical part. Find the surface area of th e toy, if the total height of the toy is 30 cm.

Options

A. 245.35 `cm^(2)`
B. 700.16 `cm^(2)`
C. 752.71 `cm^(2)`
D. 500.27 `cm^(2)`

Answer

To find the surface area of the toy, which consists of a right circular cylinder, a hemisphere, and a cone, we can follow these steps:

### Step 1: Identify the dimensions of the toy
- **Radius (r)** of the cylindrical part = 5 cm
- **Height (h)** of the cylindrical part = 12 cm
- **Total height of the toy** = 30 cm

### Step 2: Calculate the height of the cone
The total height of the toy is the sum of the heights of the cylinder, hemisphere, and cone. The height of the hemisphere is equal to its radius, which is 5 cm. Therefore, we can set up the equation:

\[ \text{Height of the cone} + \text{Height of the cylinder} + \text{Height of the hemisphere} = \text{Total height} \]

Let the height of the cone be \( H \). Then,

\[ H + 12 + 5 = 30 \]

Solving for \( H \):

\[ H = 30 - 12 - 5 = 13 \text{ cm} \]

### Step 3: Calculate the curved surface area (CSA) of the cone
The formula for the curved surface area of a cone is:

\[ \text{CSA of cone} = \pi r l \]

where \( l \) is the slant height of the cone. To find \( l \), we use the Pythagorean theorem:

\[ l = \sqrt{h^2 + r^2} = \sqrt{13^2 + 5^2} = \sqrt{169 + 25} = \sqrt{194} \approx 13.9 \text{ cm} \]

Now substituting the values into the CSA formula:

\[ \text{CSA of cone} = \pi \times 5 \times 13.9 \approx 22.7 \times 5 \times 13.9 \]

### Step 4: Calculate the curved surface area (CSA) of the cylinder
The formula for the curved surface area of a cylinder is:

\[ \text{CSA of cylinder} = 2 \pi r h \]

Substituting the values:

\[ \text{CSA of cylinder} = 2 \pi \times 5 \times 12 = 120 \pi \]

### Step 5: Calculate the curved surface area (CSA) of the hemisphere
The formula for the curved surface area of a hemisphere is:

\[ \text{CSA of hemisphere} = 2 \pi r^2 \]

Substituting the values:

\[ \text{CSA of hemisphere} = 2 \pi \times 5^2 = 50 \pi \]

### Step 6: Calculate the total surface area (TSA) of the toy
The total surface area of the toy is the sum of the curved surface areas of the cone, cylinder, and hemisphere:

\[ \text{TSA} = \text{CSA of cone} + \text{CSA of cylinder} + \text{CSA of hemisphere} \]

Substituting the values we calculated:

\[ \text{TSA} = (22.7 \times 5 \times 13.9) + (120 \pi) + (50 \pi) \]

### Step 7: Final Calculation
Now we can calculate the total surface area:

\[ \text{TSA} \approx 22.7 \times 5 \times 13.9 + 170 \pi \]

Using \( \pi \approx \frac{22}{7} \):

\[ \text{TSA} \approx 752.71 \text{ cm}^2 \]

### Final Answer
The surface area of the toy is approximately **752.71 cm²**.