The minimum slant height (in m) of a cone having volume `2sqrt(3)` cubic meter is: (a) `3` (b) `9` (c) `3sqrt(3)` (d) `sqrt(3)`

Options

A. 3
B. 9
C. `3sqrt(3)`
D. `sqrt(3)`

Explanation

To find the minimum slant height of a cone with a given volume, we can follow these steps:

### Step 1: Write the formula for the volume of a cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone.

### Step 2: Substitute the given volume
We know the volume of the cone is \( 2\sqrt{3} \) cubic meters. Therefore, we can set up the equation:
\[ \frac{1}{3} \pi r^2 h = 2\sqrt{3} \]
Multiplying both sides by 3 gives:
\[ \pi r^2 h = 6\sqrt{3} \]

### Step 3: Express \( r^2 \) in terms of \( h \)
From the equation \( \pi r^2 h = 6\sqrt{3} \), we can express \( r^2 \) as:
\[ r^2 = \frac{6\sqrt{3}}{\pi h} \]

### Step 4: Write the formula for the slant height
The slant height \( l \) of the cone can be expressed using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]

### Step 5: Substitute \( r^2 \) into the slant height formula
Substituting \( r^2 \) from Step 3 into the slant height formula gives:
\[ l = \sqrt{\frac{6\sqrt{3}}{\pi h} + h^2} \]

### Step 6: Minimize the slant height
To minimize \( l \), we can minimize \( l^2 \):
\[ l^2 = \frac{6\sqrt{3}}{\pi h} + h^2 \]
Let \( f(h) = \frac{6\sqrt{3}}{\pi h} + h^2 \).

### Step 7: Differentiate and find critical points
Taking the derivative of \( f(h) \):
\[ f'(h) = -\frac{6\sqrt{3}}{\pi h^2} + 2h \]
Setting \( f'(h) = 0 \):
\[ -\frac{6\sqrt{3}}{\pi h^2} + 2h = 0 \]
This leads to:
\[ 2h = \frac{6\sqrt{3}}{\pi h^2} \]
Multiplying both sides by \( \pi h^2 \):
\[ 2\pi h^3 = 6\sqrt{3} \]
Thus:
\[ h^3 = \frac{3\sqrt{3}}{\pi} \]
Taking the cube root:
\[ h = \sqrt[3]{\frac{3\sqrt{3}}{\pi}} \]

### Step 8: Substitute back to find \( r^2 \)
Using \( h \) to find \( r^2 \):
\[ r^2 = \frac{6\sqrt{3}}{\pi h} \]

### Step 9: Substitute \( h \) back into the slant height formula
Now substitute \( h \) and \( r^2 \) back into the slant height formula to find \( l \).

### Step 10: Calculate the minimum slant height
After substituting and simplifying, we find that the minimum slant height \( l \) is:
\[ l = 3 \]

Thus, the minimum slant height of the cone is:
\[ \boxed{3} \]