If the height of a cone is equal to its base diameter, then its slant height is

Options

A. `sqrt ( 2 r^(2) + h^(2))`
B. `r sqrt(5)`
C. `h sqrt(5)`
D. `rh sqrt(5)`

Answer

To find the slant height of a cone where the height is equal to its base diameter, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**:
- Let the height of the cone be \( H \).
- The base diameter of the cone is \( D \).
- According to the problem, \( H = D \).

2. **Relate Diameter to Radius**:
- The radius \( R \) of the cone is half of the diameter. Therefore, we can express the diameter in terms of the radius:
\[ D = 2R \]
- Since \( H = D \), we can write:
\[ H = 2R \]

3. **Use the Slant Height Formula**:
- The formula for the slant height \( L \) of a cone is given by:
\[ L^2 = R^2 + H^2 \]

4. **Substitute the Height**:
- Substitute \( H \) with \( 2R \) in the slant height formula:
\[ L^2 = R^2 + (2R)^2 \]
- Simplifying this gives:
\[ L^2 = R^2 + 4R^2 = 5R^2 \]

5. **Calculate the Slant Height**:
- To find \( L \), take the square root of both sides:
\[ L = \sqrt{5R^2} \]
- This simplifies to:
\[ L = R\sqrt{5} \]

### Final Answer:
The slant height \( L \) of the cone is \( R\sqrt{5} \).

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