A toy is in the shape of a right circular cylinder with a hemisphere on one end and a cone on the other. The height and radius of the cylindrical part are 13 cm and 5 cm respectively. The radii of the hemispherical and conical parts are the same as that of the cylindrical part. Calculate the surface area of the toy if the height of conical part is 12 cm

Options

A. `1440 cm^(2)`
B. `385 cm^(2)`
C. `1580 cm^(2)`
D. `770 cm^(2)`

Answer

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

### Step 1: Identify the dimensions
- **Radius (r)** of the cylindrical part = 5 cm
- **Height (h)** of the cylindrical part = 13 cm
- **Height (h_cone)** of the conical part = 12 cm
- **Radius (r)** of the conical part = 5 cm (same as the cylinder)

### Step 2: Calculate the slant height of the cone
The slant height (L) of the cone can be calculated using the Pythagorean theorem:
\[ L = \sqrt{r^2 + h_{cone}^2} \]
Where:
- \( r = 5 \) cm
- \( h_{cone} = 12 \) cm

Calculating:
\[ L = \sqrt{5^2 + 12^2} \]
\[ L = \sqrt{25 + 144} \]
\[ L = \sqrt{169} \]
\[ L = 13 \text{ cm} \]

### Step 3: Calculate the curved surface area of each part
1. **Curved Surface Area of the Cylinder (CSA_cylinder)**:
\[ CSA_{cylinder} = 2\pi rh \]
\[ CSA_{cylinder} = 2 \times \frac{22}{7} \times 5 \times 13 \]
\[ CSA_{cylinder} = \frac{2200}{7} \]

2. **Curved Surface Area of the Cone (CSA_cone)**:
\[ CSA_{cone} = \pi r L \]
\[ CSA_{cone} = \frac{22}{7} \times 5 \times 13 \]
\[ CSA_{cone} = \frac{1430}{7} \]

3. **Curved Surface Area of the Hemisphere (CSA_hemisphere)**:
\[ CSA_{hemisphere} = 2\pi r^2 \]
\[ CSA_{hemisphere} = 2 \times \frac{22}{7} \times 5^2 \]
\[ CSA_{hemisphere} = 2 \times \frac{22}{7} \times 25 \]
\[ CSA_{hemisphere} = \frac{1100}{7} \]

### Step 4: Calculate the total surface area of the toy
The total surface area (SA_total) of the toy is the sum of the curved surface areas of the cylinder, cone, and hemisphere:
\[ SA_{total} = CSA_{cylinder} + CSA_{cone} + CSA_{hemisphere} \]
\[ SA_{total} = \frac{2200}{7} + \frac{1430}{7} + \frac{1100}{7} \]
\[ SA_{total} = \frac{2200 + 1430 + 1100}{7} \]
\[ SA_{total} = \frac{4730}{7} \]
\[ SA_{total} = 676.43 \text{ cm}^2 \]

### Step 5: Finalize the answer
The total surface area of the toy is approximately **676.43 cm²**. However, since the options provided in the question do not match this value, we need to ensure we only consider the exposed surfaces. The surface area of the base of the hemisphere and the base of the cone are not included in the total surface area calculation.