A goldsmith desires to test the purity of a gold ornament suspected to the mixed with copper. The ornament weighs `0.25 kg` in air and is observed to displace `0.015` litre of water when immersed in it. Densities of gold and copper with respect to water are, respectively, `19.3` and `8.9`. The approximate percentage of copper in the ornament is

Options

A. `5%`
B. `10%`
C. `15%`
D. `25%`

Explanation

To determine the approximate percentage of copper in the gold ornament, we can follow these steps:

### Step 1: Calculate the volume of the ornament
The volume of water displaced by the ornament is given as `0.015 liters`. Since `1 liter = 1000 cm³`, we can convert this to cubic meters:
\[ V = 0.015 \, \text{liters} = 0.015 \times 10^{-3} \, \text{m}^3 = 1.5 \times 10^{-5} \, \text{m}^3 \]

### Step 2: Calculate the weight of the ornament
The weight of the ornament in air is given as `0.25 kg`. The weight in Newtons can be calculated as:
\[ W = m \cdot g = 0.25 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 \approx 2.4525 \, \text{N} \]

### Step 3: Use the principle of buoyancy
According to Archimedes' principle, the weight of the water displaced is equal to the weight of the ornament. The weight of the displaced water can be calculated using the volume of water displaced and the density of water (which is approximately `1000 kg/m³`):
\[ W_{\text{displaced}} = V \cdot \rho_{\text{water}} \cdot g = 1.5 \times 10^{-5} \, \text{m}^3 \cdot 1000 \, \text{kg/m}^3 \cdot 9.81 \, \text{m/s}^2 \approx 0.14715 \, \text{N} \]

### Step 4: Calculate the relative density of the ornament
The relative density (specific gravity) of the ornament can be calculated using the formula:
\[ \text{Relative Density} = \frac{W_{\text{ornament}}}{W_{\text{displaced}}} = \frac{2.4525 \, \text{N}}{0.14715 \, \text{N}} \approx 16.67 \]

### Step 5: Set up the equation for the composition of the ornament
Let \( N \) be the percentage of copper in the ornament. The relative density of the ornament can be expressed as:
\[ \text{Relative Density} = \frac{N \cdot \rho_{\text{copper}} + (100 - N) \cdot \rho_{\text{gold}}}{100} \]
Substituting the known values:
\[ 16.67 = \frac{N \cdot 8.9 + (100 - N) \cdot 19.3}{100} \]

### Step 6: Solve for \( N \)
Multiplying both sides by `100` gives:
\[ 1667 = N \cdot 8.9 + (100 - N) \cdot 19.3 \]
Expanding and rearranging:
\[ 1667 = 8.9N + 1930 - 19.3N \]
\[ 1667 - 1930 = -10.4N \]
\[ -263 = -10.4N \]
\[ N = \frac{263}{10.4} \approx 25.38 \]

### Step 7: Conclusion
The approximate percentage of copper in the ornament is about \( 25\% \).