The volume of a cone is `18 pi cm^(3)`. Find its height if height and diameter of base are same.

Answer

To find the height of the cone given that its volume is \(18 \pi \, \text{cm}^3\) and that the height and diameter of the base are the same, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Volume Formula for a Cone**:
The formula for the volume \(V\) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \(r\) is the radius of the base and \(h\) is the height of the cone.

2. **Set Up the Equation**:
We know the volume of the cone is \(18 \pi \, \text{cm}^3\). Therefore, we can set up the equation:
\[ \frac{1}{3} \pi r^2 h = 18 \pi \]

3. **Cancel Out \(\pi\)**:
Since \(\pi\) appears on both sides of the equation, we can cancel it out:
\[ \frac{1}{3} r^2 h = 18 \]

4. **Multiply Both Sides by 3**:
To eliminate the fraction, multiply both sides by 3:
\[ r^2 h = 54 \]

5. **Relate Height and Radius**:
According to the problem, the height \(h\) is equal to the diameter of the base. The diameter \(d\) is twice the radius:
\[ d = 2r \quad \text{and thus} \quad h = d = 2r \]

6. **Substitute \(h\) in the Volume Equation**:
Substitute \(h = 2r\) into the equation \(r^2 h = 54\):
\[ r^2 (2r) = 54 \]
This simplifies to:
\[ 2r^3 = 54 \]

7. **Solve for \(r^3\)**:
Divide both sides by 2:
\[ r^3 = 27 \]

8. **Find the Radius \(r\)**:
Take the cube root of both sides:
\[ r = \sqrt[3]{27} = 3 \, \text{cm} \]

9. **Calculate the Height \(h\)**:
Now, substitute \(r\) back to find \(h\):
\[ h = 2r = 2 \times 3 = 6 \, \text{cm} \]

### Final Answer:
The height of the cone is \(6 \, \text{cm}\).