Answer

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**Hint:**The thickness of the slab is determined by using the refractive index formula. By using the apparent depth of the first side, the real depth is determined, for the apparent depth of the second side, the real depth is taken as the difference of the thickness and the real depth, then the thickness is determined.

**Useful formula**

The refractive index of the glass slab is given by,

$\mu = \dfrac{{{\text{Real depth}}}}{{{\text{Apparent depth}}}}$

Where, $\mu $ is the refractive index of the glass slab.

**Complete step by step solution**

Given that,

The refractive index of the glass slab is, $\mu = 1.5$,

The apparent depth from first side is, $6\,cm$

The apparent depth from second side is, $4\,cm$

Assume that the real depth be $x$, then

The refractive index of the glass slab is given by,

$\mu = \dfrac{{{\text{Real depth}}}}{{{\text{Apparent depth}}}}\,..................\left( 1 \right)$

By substituting the real depth and apparent depth in the above equation (1), then the above equation (1) is written as,

$\mu = \dfrac{x}{6}$

By substituting the refractive index value in the above equation, then the above equation is written as,

$1.5 = \dfrac{x}{6}$

By rearranging the terms, then the above equation is written as,

$x = 1.5 \times 6$

On multiplying the terms in the above equation, then

$x = 9$

Now, using the refractive index formula for second side, then assume that the real depth is $t - x$, then

$\mu = \dfrac{{{\text{Real depth}}}}{{{\text{Apparent depth}}}}$

By substituting the refractive index, real depth and apparent depth in the above equation, then

\[1.5 = \dfrac{{t - x}}{4}\]

Now substituting the value of $x$ in the above equation, then

\[1.5 = \dfrac{{t - 9}}{4}\]

By rearranging the terms, then the above equation is written as,

\[t - 9 = 1.5 \times 4\]

On multiplying the terms in the above equation, then

\[t - 9 = 6\]

By keeping the term $t$ in one side, then

\[t = 6 + 9\]

By adding the terms in the above equation, then

$t = 15\,cm$

**Hence, the option (C) is the correct answer.**

**Note:**The refractive index is directly proportional to the real depth and the refractive index is inversely proportional to the apparent depth. As the refractive index increases when the real depth increases and the apparent depth decreases. As the refractive index decreases when the real depth decreases and the apparent depth increases.

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