To find the density of the unknown liquid, we’ll use Archimedes' principle. According to this principle, the buoyant force on an object submerged in a fluid equals the weight of the fluid displaced by the object. Here’s the step-by-step calculation:

Step 1: Calculate the Buoyant Force

The buoyant force can be found by calculating the difference between the weight of the iron in air and its apparent weight in the liquid.

$\text{Buoyant Force} = \text{Weight in Air} - \text{Apparent Weight in Liquid}$

Since weight is proportional to mass (with gravity as a constant factor, which cancels out), we can use mass directly in grams:

$\text{Buoyant Force} = 443 \, \text{g} - 257 \, \text{g} = 186 \, \text{g}$

This 186 g is the mass of the liquid displaced by the iron.

Step 2: Calculate the Volume of the Iron (Using Its Density)

Since the density of iron is given as $7.86 \, \text{g/cm}^3$, we can find the volume of the iron by dividing its mass by its density.

$\text{Volume of Iron} = \frac{\text{Mass of Iron in Air}}{\text{Density of Iron}} = \frac{443 \, \text{g}}{7.86 \, \text{g/cm}^3}$

$\text{Volume of Iron} \approx 56.38 \, \text{cm}^3$

Step 3: Calculate the Density of the Unknown Liquid

The volume of liquid displaced is equal to the volume of the iron. Therefore, the density of the liquid can be calculated as:

$\text{Density of Liquid} = \frac{\text{Mass of Displaced Liquid}}{\text{Volume of Iron}} = \frac{186 \, \text{g}}{56.38 \, \text{cm}^3}$

$\text{Density of Liquid} \approx 3.30 \, \text{g/cm}^3$

Final Answer

The density of the unknown liquid is approximately:

$\boxed{3.30 \, \text{g/cm}^3}$

Would you like further details on any part of this process?

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1. How does Archimedes' principle relate to fluid density calculations?
2. How would this calculation change if the density of iron were different?
3. What is the role of buoyant force in determining fluid density?
4. How can we experimentally determine the density of irregularly shaped objects?
5. Why does the apparent mass change when an object is submerged?

Tip: Always ensure units are consistent when performing calculations involving density and volume.