Let `A_(1), A_(2), A_(3), A_(4)` be the areas of the triangular faces of a tetrahedron, and `h_(1), h_(2), h_(3), h_(4)` be the corresponding altitudes of the tetrahedron. If the volume of tetrahedron is `1//6` cubic units, then find the minimum value of ` (A_(1) +A_(2) + A_(3) + A_(4))(h_(1)+ h_(2)+h_(3)+h_(4))` (in cubic units).

Explanation

To find the minimum value of \( (A_1 + A_2 + A_3 + A_4)(h_1 + h_2 + h_3 + h_4) \) for a tetrahedron with a given volume, we can follow these steps:

### Step 1: Understand the volume formula
The volume \( V \) of a tetrahedron can be expressed in terms of the area of its triangular faces and their corresponding altitudes. The formula for the volume related to one face is:
\[ V = \frac{1}{3} A_i h_i \]
for each face \( i \).

### Step 2: Express altitudes in terms of volume and areas
Given that the volume \( V = \frac{1}{6} \) cubic units, we can express each altitude as:
\[ h_1 = \frac{3V}{A_1}, \quad h_2 = \frac{3V}{A_2}, \quad h_3 = \frac{3V}{A_3}, \quad h_4 = \frac{3V}{A_4} \]

### Step 3: Substitute altitudes into the expression
Now, we substitute these expressions for \( h_i \) into the expression we want to minimize:
\[ (A_1 + A_2 + A_3 + A_4)(h_1 + h_2 + h_3 + h_4) = (A_1 + A_2 + A_3 + A_4) \left( \frac{3V}{A_1} + \frac{3V}{A_2} + \frac{3V}{A_3} + \frac{3V}{A_4} \right) \]

### Step 4: Factor out constants
Factoring out the constant \( 3V \):
\[ = 3V (A_1 + A_2 + A_3 + A_4) \left( \frac{1}{A_1} + \frac{1}{A_2} + \frac{1}{A_3} + \frac{1}{A_4} \right) \]

### Step 5: Apply the AM-HM inequality
Using the Arithmetic Mean - Harmonic Mean (AM-HM) inequality, we know:
\[ \frac{A_1 + A_2 + A_3 + A_4}{4} \geq \frac{4}{\frac{1}{A_1} + \frac{1}{A_2} + \frac{1}{A_3} + \frac{1}{A_4}} \]
Rearranging gives:
\[ (A_1 + A_2 + A_3 + A_4) \left( \frac{1}{A_1} + \frac{1}{A_2} + \frac{1}{A_3} + \frac{1}{A_4} \right) \geq 16 \]

### Step 6: Substitute the volume
Substituting \( V = \frac{1}{6} \):
\[ 3V \cdot 16 = 3 \cdot \frac{1}{6} \cdot 16 = 8 \]

### Conclusion
Thus, the minimum value of \( (A_1 + A_2 + A_3 + A_4)(h_1 + h_2 + h_3 + h_4) \) is:
\[ \boxed{8} \]