If V be the volume and S be the surface area of a cuboid of dimensions a, b, c, then $\frac{1}{V}$ is equal to:

Question

Accepted Answer

To determine why Option 2 is the correct answer, let's analyze the relationship between the volume $ V $ and the surface area $ S $ of a cuboid with dimensions $ a $, $ b $, and $ c $.

### Explanation for Option 2:

The volume $ V $ of a cuboid is given by:
$$ V = a \times b \times c $$

The surface area $ S $ of a cuboid is given by:
$$ S = 2(ab + bc + ca) $$

We need to find the expression for $\frac{1}{V}$. Let's evaluate Option 2:
$$ \frac{2}{S}\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) $$

Substituting the expression for $ S $:
$$ \frac{2}{2(ab + bc + ca)}\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) $$

This simplifies to:
$$ \frac{1}{ab + bc + ca}\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) $$

Further simplifying:
$$ \frac{1}{ab + bc + ca} \left(\frac{bc + ac + ab}{abc}\right) $$

This simplifies to:
$$ \frac{bc + ac + ab}{abc(ab + bc + ca)} $$

Since $ V = abc $, the expression simplifies to:
$$ \frac{1}{abc} = \frac{1}{V} $$

Thus, Option 2 correctly represents $\frac{1}{V}$.

### Explanation for why other options are incorrect:

- **Option 1: $\frac{s}{2}(a+b+c)$**

This expression does not simplify to $\frac{1}{V}$. Instead, it involves multiplying the surface area by the sum of the dimensions, which does not relate to the reciprocal of the volume.

- **Option 3: $\frac{2S}{(a+b+c)}$**

This expression involves dividing the surface area by the sum of the dimensions, which again does not simplify to $\frac{1}{V}$. It does not account for the product of the dimensions in the denominator.

- **Option 4: $2S(a+b+c)$**

This expression involves multiplying the surface area by the sum of the dimensions, which results in a completely different value that does not relate to the reciprocal of the volume.

In conclusion, Option 2 is the only expression that simplifies to $\frac{1}{V}$, making it the correct answer.