An ice cube has a bubble inside. When viewed from one side the apparent distance of the bubble is $12 \mathrm{~cm}$. When viewed from the opposite side, the apparent distance of the bubble is observed as $4 \mathrm{~cm}$. If the side of the ice cube is $24 \mathrm{~cm}$, the refractive index of the ice cube is

Let's denote the true distance of the bubble from one side of the ice cube as $x$ and the refractive index of the ice cube as $n$. We will use the formula for apparent depth, which states that the ratio of the true depth to the apparent depth is equal to the refractive index: <br/><br/> $n = \frac{\text{True depth}}{\text{Apparent depth}}$ <br/><br/> When viewing the bubble from one side, the true depth is $x$ and the apparent depth is $12 \mathrm{~cm}$. Using the formula: <br/><br/> $n = \frac{x}{12}$ <br/><br/> When viewing the bubble from the opposite side, the true depth is $24 - x$ (since the side of the ice cube is $24 \mathrm{~cm}$) and the apparent depth is $4 \mathrm{~cm}$. Using the formula: <br/><br/> $n = \frac{24 - x}{4}$ <br/><br/> Now we have a system of two equations with two variables: <br/><br/> 1) $n = \frac{x}{12}$<br/><br/> 2) $n = \frac{24 - x}{4}$ <br/><br/> We can solve this system by setting the two expressions for $n$ equal to each other: <br/><br/> $\frac{x}{12} = \frac{24 - x}{4}$ <br/><br/> To solve for $x$, first multiply both sides by $12$: <br/><br/> $x = 3(24 - x)$ <br/><br/> $x = 72 - 3x$ <br/><br/> Add $3x$ to both sides: <br/><br/> $4x = 72$ <br/><br/> Divide by $4$: <br/><br/> $x = 18$ <br/><br/> Now that we have the value of $x$, we can find the refractive index $n$ using either equation 1 or 2. Using equation 1: <br/><br/> $n = \frac{18}{12} = \frac{3}{2}$ <br/><br/> Therefore, the refractive index of the ice cube is $\frac{3}{2}$.