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 \( k \) given that the product of the height and the square of the radius of the greatest cone obtained by rotating a right-angle triangle of hypotenuse 2 meters about one of its sides is \( \frac{k}{3\sqrt{3}} \).

### Step-by-Step Solution:

1. **Understanding the Triangle**:
We have a right-angle triangle with a hypotenuse of 2 meters. Let the sides be \( a \), \( b \), and \( c \) (where \( c \) is the hypotenuse). According to the Pythagorean theorem, we have:
\[ a^2 + b^2 = c^2 \]
Here, \( c = 2 \), so:
\[ a^2 + b^2 = 2^2 = 4 \]

2. **Setting Up the Cone**:
When we rotate the triangle about one of its sides (let's say side \( AB \)), we form a cone where:
- Height \( h = a \)
- Radius \( r = b \)

3. **Using Pythagorean Theorem**:
From the triangle, we can express \( r \) in terms of \( h \):
\[ r^2 + h^2 = 4 \implies r^2 = 4 - h^2 \]

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

5. **Maximizing the Volume**:
To find the maximum volume, we differentiate \( V \) with respect to \( h \):
\[ \frac{dV}{dh} = \frac{1}{3} \pi (4 - 3h^2) \]
Setting the derivative to zero for maximization:
\[ 4 - 3h^2 = 0 \implies 3h^2 = 4 \implies h^2 = \frac{4}{3} \implies h = \frac{2}{\sqrt{3}} \]

6. **Finding the Radius**:
Now substituting \( h \) back to find \( r^2 \):
\[ r^2 = 4 - h^2 = 4 - \frac{4}{3} = \frac{12}{3} - \frac{4}{3} = \frac{8}{3} \]

7. **Calculating the Product**:
The product of height and square of the radius is:
\[ h \cdot r^2 = \left(\frac{2}{\sqrt{3}}\right) \cdot \left(\frac{8}{3}\right) = \frac{16}{3\sqrt{3}} \]

8. **Finding \( k \)**:
According to the problem, this product equals \( \frac{k}{3\sqrt{3}} \):
\[ \frac{16}{3\sqrt{3}} = \frac{k}{3\sqrt{3}} \]
Therefore, \( k = 16 \).

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