The numerical value of product of the sides of a triangle is 512 units . Find the minimum possible perimeter of the triangle ( in units ).

Options

A. 18
B. 24
C. 30
D. 22

Answer

To find the minimum possible perimeter of a triangle given that the product of its sides is 512 units, we can use the concept of the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Let the sides of the triangle be \( a \), \( b \), and \( c \)**:
We know that \( a \times b \times c = 512 \).

2. **Using AM-GM Inequality**:
According to the AM-GM inequality, for any non-negative numbers, the arithmetic mean is greater than or equal to the geometric mean. Therefore, we have:
\[ \frac{a + b + c}{3} \geq \sqrt[3]{abc} \]

3. **Substituting the product**:
Since \( abc = 512 \), we can substitute this into the inequality:
\[ \frac{a + b + c}{3} \geq \sqrt[3]{512} \]

4. **Calculating the cube root**:
We calculate \( \sqrt[3]{512} \):
\[ \sqrt[3]{512} = 8 \]
(since \( 8 \times 8 \times 8 = 512 \)).

5. **Finding the minimum perimeter**:
Now substituting back into the inequality:
\[ \frac{a + b + c}{3} \geq 8 \]
Multiplying both sides by 3 gives:
\[ a + b + c \geq 24 \]
Therefore, the minimum possible perimeter \( P \) of the triangle is:
\[ P \geq 24 \text{ units} \]

6. **Condition for equality**:
The equality in the AM-GM inequality holds when \( a = b = c \). Thus, if we take \( a = b = c \), we can find the specific values of the sides:
\[ a = b = c = \sqrt[3]{512} = 8 \]

7. **Conclusion**:
Therefore, the minimum possible perimeter of the triangle is:
\[ \text{Minimum Perimeter} = 8 + 8 + 8 = 24 \text{ units} \]