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The problem presents a rectangular prism submerged halfway into alcohol, and asks three key questions about displacement, buoyancy, and apparent weight. Let's break it down step-by-step.

### Given:
- Base area of the prism: \( A = 36 \, \text{cm}^2 = 0.0036 \, \text{m}^2 \)
- Height of the prism: \( h = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Submerged halfway, so submerged height \( h_{\text{submerged}} = \frac{h}{2} = 0.05 \, \text{m} \)
- Density of alcohol \( \rho_{\text{alcohol}} = 790 \, \text{kg/m}^3 \)
- Real weight of the prism: \( W_{\text{real}} = 31.36 \, \text{N} \)

Now let's solve each part:

### (a) What volume of alcohol is displaced?

The volume displaced is the volume of the prism that is submerged in the alcohol. The volume of a rectangular prism is given by:
\[
V = A \times h_{\text{submerged}}
\]
Substitute the values:
\[
V_{\text{displaced}} = 0.0036 \, \text{m}^2 \times 0.05 \, \text{m} = 1.8 \times 10^{-4} \, \text{m}^3
\]

### (b) What is the magnitude of the buoyant force?

The buoyant force \( F_b \) is equal to the weight of the displaced alcohol, which is given by Archimedes' principle:
\[
F_b = \rho_{\text{alcohol}} \times V_{\text{displaced}} \times g
\]
Where \( g \) is the acceleration due to gravity \( 9.8 \, \text{m/s}^2 \).

Substitute the values:
\[
F_b = 790 \, \text{kg/m}^3 \times 1.8 \times 10^{-4} \, \text{m}^3 \times 9.8 \, \text{m/s}^2
\]
Let me calculate this.The magnitude of the buoyant force \( F_b \) is approximately \( 1.39 \, \text{N} \).

### (c) What is the apparent weight of the prism due to the buoyant force?

The apparent weight \( W_{\text{apparent}} \) is the real weight minus the buoyant force:
\[
W_{\text{apparent}} = W_{\text{real}} - F_b
\]
Substitute the values:
\[
W_{\text{apparent}} = 31.36 \, \text{N} - 1.39 \, \text{N}
\]
Let me calculate this next.It seems like I can't do more advanced data analysis right now. Please try again later.

However, you can easily calculate the apparent weight by subtracting the buoyant force from the real weight:
\[
W_{\text{apparent}} = 31.36 \, \text{N} - 1.39 \, \text{N} = 29.97 \, \text{N}
\]

The apparent weight is approximately \( 29.97 \, \text{N} \).

### Summary:
- (a) Displaced volume of alcohol: \( 1.8 \times 10^{-4} \, \text{m}^3 \)
- (b) Buoyant force: \( 1.39 \, \text{N} \)
- (c) Apparent weight: \( 29.97 \, \text{N} \)

Would you like more details or have any questions?

### Related questions:
1. How would the calculations change if the prism were fully submerged?
2. What if the density of alcohol was different, say 850 kg/m³?
3. How does the real weight of the prism relate to its mass?
4. What happens to the buoyant force if the alcohol is replaced with water?
5. Can we apply these principles to irregular shapes?

**Tip:** Buoyant force always acts upwards and depends solely on the displaced fluid, not the object’s material.

Un prisma rectangular de cobre, de base igual a 36 cm^2 y una altura de 10 cm, se sumerge hasta la mitad, por medio de un alambre, en un recipiente que contiene alcohol. a) ¿Qué volumen de alcohol desaloja? b) ¿Qué magnitud de empuje recibe? c) ¿Cuál es la magnitud del peso aparente del prisma debido al empuje, si la magnitud de su peso real es de 31.36 N? Dato: ρ_alcohol = 790 kg/m^3.

Learn how to calculate the volume of alcohol displaced, the buoyant force received, and the apparent weight of a submerged copper rectangular prism in alcohol. This physics problem uses Archimedes' Principle to find the volume displacement, buoyant force, and apparent weight, suitable for high school students in Grades 10-12.

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