A vessel contains water to a depth of 4 cm. Floating on the water is a layer of oil (n = 1.61) 5 cm deep. What is the apparent distance of the bottom of the vessel below the surface of oil?

To find the apparent distance of the bottom of the vessel below the surface of the oil, we need to account for the refraction of light as it passes through the two different layers: water and oil.

### Step-by-Step Calculation:

1. **Real depth of the water layer:**
   - Depth of water, \( d_{\text{water}} = 4 \text{ cm} \).
   - Refractive index of water, \( n_{\text{water}} = 1.33 \).

2. **Real depth of the oil layer:**
   - Depth of oil, \( d_{\text{oil}} = 5 \text{ cm} \).
   - Refractive index of oil, \( n_{\text{oil}} = 1.61 \).

The apparent depth can be found by using the following formula for apparent depth when light passes through a medium:
\[
\text{Apparent depth} = \frac{\text{Real depth}}{\text{Refractive index}}
\]
We apply this separately for both the water and the oil layers.

3. **Apparent depth of the water layer (as seen from within the oil):**
   \[
   d'_{\text{water}} = \frac{d_{\text{water}}}{n_{\text{water}}} = \frac{4}{1.33} \approx 3.01 \text{ cm}
   \]

4. **Apparent depth of the oil layer (as seen from the air):**
   \[
   d'_{\text{oil}} = \frac{d_{\text{oil}}}{n_{\text{oil}}} = \frac{5}{1.61} \approx 3.11 \text{ cm}
   \]

### Total apparent depth:
The total apparent depth of the bottom of the vessel, as seen from the surface of the oil, is the sum of the apparent depths of the two layers:
\[
d'_{\text{total}} = d'_{\text{oil}} + d'_{\text{water}} \approx 3.11 + 3.01 = 6.12 \text{ cm}
\]

Thus, the apparent distance of the bottom of the vessel below the surface of the oil is approximately **6.12 cm**.

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Would you like any further details or have any questions?

### Related Questions:
1. How does the refractive index of the medium affect the apparent depth?
2. What would happen to the apparent depth if the oil had a higher refractive index?
3. Can we generalize the formula for multiple layers of different refractive indices?
4. How would the apparent depth change if the oil layer were thinner?
5. What happens if the viewing medium (instead of air) has a refractive index?

### Tip:
Always remember that light bends toward the normal when it enters a medium with a higher refractive index, and away from the normal when it enters a medium with a lower refractive index.

Explore how to calculate the apparent depth of a vessel with 4 cm of water and 5 cm of oil (n=1.61). Understand refraction principles, including how light behaves through different media. This problem is ideal for high school students learning optics and refraction. Step-by-step solution included.

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