A rocket is in the form of a right circular cylinder closed at the lower end and surmounted by a cone with the same radius as that of cylinder. The diameter and height of cylinder are 6 cm and 12 cm, respectively. If the slant height of the conical portion is 5 cm, then find the total surface area and volume of the rocket, (use `pi` = 3.14)

Answer

To solve the problem of finding the total surface area and volume of the rocket, we will break it down into steps.

### Step 1: Find the radius of the cylinder and cone
Given:
- Diameter of the cylinder = 6 cm
- Radius (r) = Diameter / 2 = 6 cm / 2 = 3 cm

### Step 2: Identify the height of the cylinder and the slant height of the cone
Given:
- Height of the cylinder (H) = 12 cm
- Slant height of the cone (L) = 5 cm

### Step 3: Find the height of the cone
We can use the Pythagorean theorem to find the height (h) of the cone. The relationship is given by:
\[ L^2 = h^2 + r^2 \]
Substituting the known values:
\[ 5^2 = h^2 + 3^2 \]
\[ 25 = h^2 + 9 \]
\[ h^2 = 25 - 9 = 16 \]
\[ h = \sqrt{16} = 4 \text{ cm} \]

### Step 4: Calculate the volume of the rocket
The volume of the rocket is the sum of the volume of the cylinder and the volume of the cone.

**Volume of the cylinder (V_cylinder)**:
\[ V_{cylinder} = \pi r^2 H \]
Substituting the values:
\[ V_{cylinder} = 3.14 \times (3^2) \times 12 \]
\[ V_{cylinder} = 3.14 \times 9 \times 12 = 3.14 \times 108 = 339.12 \text{ cm}^3 \]

**Volume of the cone (V_cone)**:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Substituting the values:
\[ V_{cone} = \frac{1}{3} \times 3.14 \times (3^2) \times 4 \]
\[ V_{cone} = \frac{1}{3} \times 3.14 \times 9 \times 4 = \frac{1}{3} \times 3.14 \times 36 = 3.14 \times 12 = 37.68 \text{ cm}^3 \]

**Total Volume (V_total)**:
\[ V_{total} = V_{cylinder} + V_{cone} \]
\[ V_{total} = 339.12 + 37.68 = 376.8 \text{ cm}^3 \]

### Step 5: Calculate the total surface area of the rocket
The total surface area (TSA) of the rocket consists of the curved surface area (CSA) of the cylinder, the curved surface area of the cone, and the base area of the cylinder.

**Curved Surface Area of the Cylinder (CSA_cylinder)**:
\[ CSA_{cylinder} = 2 \pi r H \]
Substituting the values:
\[ CSA_{cylinder} = 2 \times 3.14 \times 3 \times 12 \]
\[ CSA_{cylinder} = 2 \times 3.14 \times 36 = 226.08 \text{ cm}^2 \]

**Curved Surface Area of the Cone (CSA_cone)**:
\[ CSA_{cone} = \pi r L \]
Substituting the values:
\[ CSA_{cone} = 3.14 \times 3 \times 5 = 47.1 \text{ cm}^2 \]

**Base Area of the Cylinder**:
\[ Base_{cylinder} = \pi r^2 \]
Substituting the values:
\[ Base_{cylinder} = 3.14 \times (3^2) = 3.14 \times 9 = 28.26 \text{ cm}^2 \]

**Total Surface Area (TSA)**:
\[ TSA = CSA_{cylinder} + CSA_{cone} + Base_{cylinder} \]
\[ TSA = 226.08 + 47.1 + 28.26 = 301.44 \text{ cm}^2 \]

### Final Results
- Total Surface Area of the rocket = 301.44 cm²
- Volume of the rocket = 376.8 cm³