Let's break down the problem step by step.

Given:

• Density of iron = $7.87 \, \text{g/cm}^3$
• Volume of the iron sphere = $30 \, \text{cm}^3$
• Mass of displaced liquid = $50 \, \text{g}$

a. Calculate the volume of the displaced liquid

According to Archimedes' principle, when an object is fully submerged in a liquid, the volume of the liquid displaced is equal to the volume of the object submerged.

Thus, since the iron sphere is fully submerged, the volume of the displaced liquid is the same as the volume of the sphere:

$\text{Volume of displaced liquid} = 30 \, \text{cm}^3$

b. Determine the density of the mystery liquid

The formula for density is:

$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

We are given:

• Mass of the displaced liquid = $50 \, \text{g}$
• Volume of the displaced liquid = $30 \, \text{cm}^3$

Now, we can calculate the density:

$\text{Density of mystery liquid} = \frac{50 \, \text{g}}{30 \, \text{cm}^3} = 1.67 \, \text{g/cm}^3$

Final Answers:

• a. The volume of the displaced liquid is $30 \, \text{cm}^3$.
• b. The density of the mystery liquid is $1.67 \, \text{g/cm}^3$.

Do you have any questions or need further clarification?

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Here are some follow-up questions to deepen understanding:

1. How would the calculation change if the sphere were only partially submerged?
2. What is Archimedes' principle, and how does it apply to this problem?
3. How would you find the mass of the iron sphere from the given data?
4. If the density of the liquid were less than the density of the sphere, would the sphere float or sink?
5. How can this experiment help identify the mystery liquid based on its density?

Tip: Always ensure that the units are consistent when applying physical formulas (e.g., volume in cm³ and mass in grams when calculating density in g/cm³).