The diameter of the base of a right circular cone is 4 cm and its height `2sqrt(3) cm`. The slant height of the cone is

Options

A. 5 cm
B. 4 cm and 8 cm
C. `2sqrt(3) cm`
D. 3 cm

Answer

To find the slant height of a right circular cone, we can use the Pythagorean theorem. The slant height (l), radius (r), and height (h) of the cone are related by the equation:

\[ l^2 = r^2 + h^2 \]

### Step-by-step solution:

1. **Identify the given values:**
- Diameter of the base of the cone = 4 cm
- Height of the cone (h) = \( 2\sqrt{3} \) cm

2. **Calculate the radius of the base:**
- Radius (r) = Diameter / 2
- \( r = 4 \, \text{cm} / 2 = 2 \, \text{cm} \)

3. **Use the Pythagorean theorem to find the slant height (l):**
- According to the theorem:
\[ l^2 = r^2 + h^2 \]
- Substitute the values of r and h:
\[ l^2 = (2 \, \text{cm})^2 + (2\sqrt{3} \, \text{cm})^2 \]

4. **Calculate \( r^2 \) and \( h^2 \):**
- \( r^2 = 2^2 = 4 \, \text{cm}^2 \)
- \( h^2 = (2\sqrt{3})^2 = 4 \cdot 3 = 12 \, \text{cm}^2 \)

5. **Add \( r^2 \) and \( h^2 \):**
- \( l^2 = 4 \, \text{cm}^2 + 12 \, \text{cm}^2 = 16 \, \text{cm}^2 \)

6. **Take the square root to find the slant height (l):**
- \( l = \sqrt{16 \, \text{cm}^2} = 4 \, \text{cm} \)

### Final Answer:
The slant height of the cone is \( 4 \, \text{cm} \).

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