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To solve this problem, let's first restate it in English:

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A solid and homogeneous sphere floats in a liquid, with the volume of the immersed part being three times the volume of the part that remains above the surface of the fluid. Then the density of the sphere is equal to:

A. half the density of the fluid  
B. four-thirds the density of the fluid  
C. three-quarters the density of the fluid  
D. one-third the density of the fluid  
E. two-thirds the density of the fluid

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**Solution:**

To determine the density of the sphere (\(\rho_{\text{sphere}}\)) in relation to the density of the fluid (\(\rho_{\text{fluid}}\)), we use the principle of buoyancy, which states that the weight of the fluid displaced by the submerged part of the sphere is equal to the weight of the sphere.

1. **Let \(V\) be the total volume of the sphere.**
2. **Let \(V_{\text{sub}}\) be the volume of the submerged part.**
3. **Let \(V_{\text{above}}\) be the volume of the part above the fluid.**

Given that \(V_{\text{sub}} = 3 \cdot V_{\text{above}}\):

\[ V_{\text{sub}} + V_{\text{above}} = V \]
\[ V_{\text{sub}} + \frac{V_{\text{sub}}}{3} = V \]
\[ V_{\text{sub}} \left(1 + \frac{1}{3}\right) = V \]
\[ V_{\text{sub}} \cdot \frac{4}{3} = V \]
\[ V_{\text{sub}} = \frac{3}{4} V \]

The buoyant force equals the weight of the submerged volume of fluid, which can be written as:

\[ \rho_{\text{fluid}} \cdot V_{\text{sub}} \cdot g = \rho_{\text{sphere}} \cdot V \cdot g \]

Simplifying:

\[ \rho_{\text{fluid}} \cdot \frac{3}{4} V = \rho_{\text{sphere}} \cdot V \]

\[ \rho_{\text{fluid}} \cdot \frac{3}{4} = \rho_{\text{sphere}} \]

\[ \rho_{\text{sphere}} = \frac{3}{4} \rho_{\text{fluid}} \]

So, the density of the sphere is three-quarters of the density of the fluid.

**Correct answer:**

C. three-quarters the density of the fluid

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Would you like more details or have any other questions? 

Here are some more questions you might be interested in:

1. How to calculate the volume of a sphere?
2. What is Archimedes' principle and how is it applied in physics?
3. How do you determine the density of an irregular object?
4. What is the relationship between mass, volume, and density?
5. How does the density of different fluids affect buoyancy?

**Tip:** Remember, buoyancy is directly related to the density of both the object and the fluid. This principle is crucial for understanding why objects float or sink.

Explore how to determine the density of a sphere floating in a liquid using the principles of buoyancy and Archimedes' principle. This problem involves calculating densities and volumes related to fluid mechanics and is suitable for high school students.

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