If the product of height and square of the radius of the greatest cone obtained by rotating a right - angle triangle of hypotenuse 2 meters about a side is `(k)/(3sqrt3)`, then k is equal to

Answer

To solve the problem, we need to find the value of \( k \) in the expression given for the product of the height and square of the radius of the greatest cone obtained by rotating a right-angle triangle with a hypotenuse of 2 meters about one of its sides.

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a right-angle triangle with a hypotenuse of 2 meters. Let the vertices be \( A \), \( B \), and \( O \) where \( O \) is the right angle.
- We rotate the triangle about one of its sides (let's say side \( AO \)) to form a cone.

2. **Identify the Variables**:
- Let \( AO = x \) (the height of the cone).
- Let \( OC = r \) (the radius of the cone).
- By the Pythagorean theorem, since \( AC \) is the hypotenuse:
\[ AO^2 + OC^2 = AC^2 \implies x^2 + r^2 = 2^2 \implies x^2 + r^2 = 4 \]

3. **Express \( r^2 \) in terms of \( x \)**:
- Rearranging the equation gives:
\[ r^2 = 4 - x^2 \]

4. **Volume of the Cone**:
- The volume \( V \) of the cone is given by:
\[ V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 x \]
- Substituting \( r^2 \):
\[ V = \frac{1}{3} \pi (4 - x^2)x = \frac{1}{3} \pi (4x - x^3) \]

5. **Find the Maximum Volume**:
- To find the maximum volume, we differentiate \( V \) with respect to \( x \):
\[ \frac{dV}{dx} = \frac{1}{3} \pi (4 - 3x^2) \]
- Set the derivative equal to zero to find critical points:
\[ 4 - 3x^2 = 0 \implies 3x^2 = 4 \implies x^2 = \frac{4}{3} \implies x = \frac{2}{\sqrt{3}} \]

6. **Calculate \( r^2 \)**:
- Substitute \( x = \frac{2}{\sqrt{3}} \) into the equation for \( r^2 \):
\[ r^2 = 4 - x^2 = 4 - \frac{4}{3} = \frac{12}{3} - \frac{4}{3} = \frac{8}{3} \]

7. **Calculate the Product**:
- Now, calculate the product of height and square of radius:
\[ x \cdot r^2 = \left(\frac{2}{\sqrt{3}}\right) \cdot \left(\frac{8}{3}\right) = \frac{16}{3\sqrt{3}} \]

8. **Identify \( k \)**:
- According to the problem, this product is given as \( \frac{k}{3\sqrt{3}} \). Thus, we have:
\[ \frac{16}{3\sqrt{3}} = \frac{k}{3\sqrt{3}} \implies k = 16 \]

### Final Answer:
The value of \( k \) is \( \boxed{16} \).