DABC be a tetrahedron such that AD is perpendicular to the base ABC and `angle ABC=30^(@)`. The volume of tetrahedron is 18. if value of `AB+BC+AD` is minimum, then the length of AC is

Options

A. `6sqrt(2-sqrt(3))`
B. `3(sqrt(6)-sqrt(2))`
C. `6sqrt(2+sqrt(3))`
D. `3(sqrt(6)+sqrt(2))`.

Answer

To solve the problem step by step, we will break down the information given and use the properties of tetrahedrons and triangles.

### Step 1: Understand the Volume of the Tetrahedron
The volume \( V \) of a tetrahedron can be calculated using the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
In this case, the height is \( AD \) and the base area can be calculated using the triangle \( ABC \).

### Step 2: Calculate the Area of Triangle ABC
Given that \( \angle ABC = 30^\circ \), we can express the area of triangle \( ABC \) as:
\[ \text{Area}_{ABC} = \frac{1}{2} \times AB \times BC \times \sin(30^\circ) = \frac{1}{2} \times AB \times BC \times \frac{1}{2} = \frac{1}{4} AB \times BC \]

### Step 3: Substitute into the Volume Formula
Substituting the area into the volume formula:
\[ V = \frac{1}{3} \times \frac{1}{4} AB \times BC \times AD \]
Given that the volume \( V = 18 \), we have:
\[ 18 = \frac{1}{12} AB \times BC \times AD \]
Multiplying both sides by 12 gives:
\[ 216 = AB \times BC \times AD \]

### Step 4: Apply the AM-GM Inequality
To minimize \( AB + BC + AD \), we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{AB + BC + AD}{3} \geq \sqrt[3]{AB \times BC \times AD} \]
Substituting \( AB \times BC \times AD = 216 \):
\[ \frac{AB + BC + AD}{3} \geq \sqrt[3]{216} = 6 \]
Thus, we have:
\[ AB + BC + AD \geq 18 \]
The minimum occurs when \( AB = BC = AD = 6 \).

### Step 5: Find the Length of AC
Using the cosine rule in triangle \( ABC \):
\[ AC^2 = AB^2 + BC^2 - 2 \times AB \times BC \times \cos(30^\circ) \]
Substituting \( AB = 6 \) and \( BC = 6 \):
\[ AC^2 = 6^2 + 6^2 - 2 \times 6 \times 6 \times \frac{\sqrt{3}}{2} \]
Calculating:
\[ AC^2 = 36 + 36 - 36\sqrt{3} \]
\[ AC^2 = 72 - 36\sqrt{3} \]
Taking the square root:
\[ AC = \sqrt{72 - 36\sqrt{3}} \]

### Final Answer
Thus, the length of \( AC \) is:
\[ AC = 6\sqrt{2 - \sqrt{3}} \]