Find the height of largest right circular cone that can be cut out of a cube whose volume is `729 cm^3`

Answer

To find the height of the largest right circular cone that can be cut out of a cube with a volume of \(729 \, \text{cm}^3\), follow these steps:

### Step 1: Find the edge length of the cube
The volume \(V\) of a cube is given by the formula:
\[ V = a^3 \]
where \(a\) is the edge length of the cube. Given that the volume is \(729 \, \text{cm}^3\), we can set up the equation:
\[ a^3 = 729 \]

### Step 2: Calculate the cube root
To find \(a\), we take the cube root of both sides:
\[ a = \sqrt[3]{729} \]
Calculating the cube root, we find:
\[ a = 9 \, \text{cm} \]

### Step 3: Relate the cone's dimensions to the cube
In a right circular cone that fits inside the cube, the diameter of the base of the cone is equal to the edge length of the cube. Therefore, the radius \(r\) of the cone is:
\[ r = \frac{a}{2} = \frac{9}{2} = 4.5 \, \text{cm} \]
The height \(H\) of the cone is equal to the edge length of the cube:
\[ H = a = 9 \, \text{cm} \]

### Step 4: Conclusion
Thus, the height of the largest right circular cone that can be cut out of the cube is:
\[ \boxed{9 \, \text{cm}} \]

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