A crown made of gold and copper weights 210g in air and 198 g in water the weight of gold in crown is
[Given, Density of gold `=19.3g//cm^(2)` and density of copper `=8.5g//cm^(3)`]

Options

A. 93 g
B. 100g
C. 150 g
D. 193g

Answer

To solve the problem of finding the weight of gold in the crown made of gold and copper, we will follow these steps:

### Step 1: Define Variables
Let:
- \( V_g \) = Volume of gold in the crown (in cm³)
- \( V_c \) = Volume of copper in the crown (in cm³)

### Step 2: Set Up the First Equation
The total weight of the crown in air is given as 210 g. We can express this in terms of the volumes and densities of gold and copper:
\[ \text{Weight in air} = \text{Weight of gold} + \text{Weight of copper} \]
Using the densities:
\[ 210 = (19.3 \times V_g) + (8.5 \times V_c) \tag{1} \]

### Step 3: Set Up the Second Equation
The weight of the crown in water is given as 198 g. The loss of weight when submerged in water is due to the buoyant force, which can be calculated as:
\[ \text{Loss of weight} = \text{Weight in air} - \text{Weight in water} = 210 - 198 = 12 \text{ g} \]
The buoyant force is equal to the weight of the water displaced, which is equal to the volume of the crown submerged multiplied by the density of water (1 g/cm³):
\[ \text{Buoyant force} = V_g + V_c = 12 \tag{2} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( 19.3 V_g + 8.5 V_c = 210 \)
2. \( V_g + V_c = 12 \)

From equation (2), we can express \( V_c \) in terms of \( V_g \):
\[ V_c = 12 - V_g \]

### Step 5: Substitute and Solve for \( V_g \)
Substituting \( V_c \) into equation (1):
\[ 19.3 V_g + 8.5(12 - V_g) = 210 \]
Expanding this:
\[ 19.3 V_g + 102 - 8.5 V_g = 210 \]
Combining like terms:
\[ (19.3 - 8.5)V_g + 102 = 210 \]
\[ 10.8 V_g + 102 = 210 \]
Subtracting 102 from both sides:
\[ 10.8 V_g = 108 \]
Dividing by 10.8:
\[ V_g = 10 \text{ cm}^3 \]

### Step 6: Find \( V_c \)
Using \( V_g \) to find \( V_c \):
\[ V_c = 12 - V_g = 12 - 10 = 2 \text{ cm}^3 \]

### Step 7: Calculate the Weight of Gold
Now we can calculate the weight of the gold:
\[ \text{Weight of gold} = V_g \times \text{Density of gold} = 10 \times 19.3 = 193 \text{ g} \]

### Final Answer
The weight of gold in the crown is **193 g**.

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