A sample of metal weights 210 grams in air 180 grams n water and 120 grams in an unknown liquid then

Options

A. the density of metal is `3g//cm^(3)`
B. the density of metal is `7g//cm^(3)`
C. density of metal is 4 times the density of the unknown liquid.
D. the metal will float in water.

Answer

To solve the problem, we need to determine the density of the metal and the unknown liquid based on the given weights in air, water, and the unknown liquid.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Weight of the metal in air (W_air) = 210 grams
- Weight of the metal in water (W_water) = 180 grams
- Weight of the metal in the unknown liquid (W_unknown) = 120 grams

2. **Calculate the Buoyant Force in Water:**
The buoyant force (B) acting on the metal when it is submerged in water can be calculated as:
\[ B = W_{\text{air}} - W_{\text{water}} = 210 \, \text{grams} - 180 \, \text{grams} = 30 \, \text{grams} \]

3. **Calculate the Volume of the Metal:**
The buoyant force is equal to the weight of the water displaced by the metal. Since the density of water (ρ_water) is 1 gram/cm³, we can find the volume (V) of the metal:
\[ B = V \cdot \rho_{\text{water}} \cdot g \]
Thus, we have:
\[ V = \frac{B}{\rho_{\text{water}} \cdot g} = \frac{30 \, \text{grams}}{1 \, \text{gram/cm}^3} = 30 \, \text{cm}^3 \]

4. **Calculate the Density of the Metal:**
The density (ρ_metal) of the metal can be calculated using its mass and volume:
\[ \rho_{\text{metal}} = \frac{W_{\text{air}}}{V} = \frac{210 \, \text{grams}}{30 \, \text{cm}^3} = 7 \, \text{grams/cm}^3 \]

5. **Determine the Density of the Unknown Liquid:**
Now, we can find the buoyant force when the metal is submerged in the unknown liquid:
\[ B_{\text{unknown}} = W_{\text{air}} - W_{\text{unknown}} = 210 \, \text{grams} - 120 \, \text{grams} = 90 \, \text{grams} \]
Since the buoyant force is also equal to the weight of the liquid displaced:
\[ B_{\text{unknown}} = V \cdot \rho_{\text{unknown}} \cdot g \]
We already know the volume of the metal (30 cm³), so:
\[ 90 \, \text{grams} = 30 \, \text{cm}^3 \cdot \rho_{\text{unknown}} \cdot g \]
Thus, we can find the density of the unknown liquid:
\[ \rho_{\text{unknown}} = \frac{90 \, \text{grams}}{30 \, \text{cm}^3} = 3 \, \text{grams/cm}^3 \]

### Summary of Results:
- Density of the metal = 7 grams/cm³
- Density of the unknown liquid = 3 grams/cm³