An ornament weighing `36 g` in air weighs only `34 g` in water. Assuming that some copper is mixed with gold to prepare the ornament, find the amount of copper in it. Specific gravity of gold is `19.3` and that of copper is `8.9`

Options

A. `2.2g`
B. `4.4g`
C. `1.1g`
D. `3.6g`

Explanation

To solve the problem of finding the amount of copper in the ornament, we can follow these steps:

### Step 1: Understand the Problem
The ornament weighs 36 g in air and 34 g in water. The loss of weight in water indicates the volume of water displaced, which is equal to the volume of the ornament.

### Step 2: Calculate the Loss of Weight
The loss of weight when the ornament is submerged in water can be calculated as:
\[ \text{Loss of weight} = \text{Weight in air} - \text{Weight in water} = 36 \, \text{g} - 34 \, \text{g} = 2 \, \text{g} \]

### Step 3: Relate Loss of Weight to Volume
According to Archimedes' principle, the loss of weight in water is equal to the weight of the water displaced. Since the density of water is \(1 \, \text{g/cm}^3\), the volume of water displaced (which is equal to the volume of the ornament) is:
\[ \text{Volume of ornament} = \text{Loss of weight} = 2 \, \text{cm}^3 \]

### Step 4: Set Up the Equation for Volume
Let \(M\) be the mass of copper in the ornament. Then the mass of gold in the ornament is \(36 - M\). The volumes of copper and gold can be expressed as:
\[ V_{\text{copper}} = \frac{M}{\text{Specific Gravity of Copper}} = \frac{M}{8.9} \]
\[ V_{\text{gold}} = \frac{36 - M}{\text{Specific Gravity of Gold}} = \frac{36 - M}{19.3} \]

### Step 5: Write the Volume Equation
The total volume of the ornament is the sum of the volumes of copper and gold, which equals the volume of water displaced:
\[ V_{\text{copper}} + V_{\text{gold}} = 2 \]
Substituting the expressions for volumes:
\[ \frac{M}{8.9} + \frac{36 - M}{19.3} = 2 \]

### Step 6: Solve the Equation
To solve for \(M\), we first find a common denominator, which is \(8.9 \times 19.3\):
\[ \frac{19.3M + 8.9(36 - M)}{8.9 \times 19.3} = 2 \]
Multiplying both sides by \(8.9 \times 19.3\):
\[ 19.3M + 8.9(36 - M) = 2 \times 8.9 \times 19.3 \]
Expanding this gives:
\[ 19.3M + 320.4 - 8.9M = 2 \times 171.37 \]
Combining like terms:
\[ (19.3 - 8.9)M + 320.4 = 342.74 \]
This simplifies to:
\[ 10.4M = 342.74 - 320.4 \]
\[ 10.4M = 22.34 \]
Now, dividing both sides by \(10.4\):
\[ M = \frac{22.34}{10.4} \approx 2.14 \, \text{g} \]

### Step 7: Conclusion
Thus, the amount of copper in the ornament is approximately \(2.14 \, \text{g}\).

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