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 find the weight of gold in the crown, we can 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 Equations
From the problem, we know:
1. The weight of the crown in air is 210 g:
\[ \rho_g V_g + \rho_c V_c = 210 \quad \text{(1)} \]
where \( \rho_g = 19.3 \, \text{g/cm}^3 \) (density of gold) and \( \rho_c = 8.5 \, \text{g/cm}^3 \) (density of copper).

2. The weight of the crown in water is 198 g. The buoyant force (thrust) acting on the crown when submerged in water reduces its weight:
\[ \rho_g V_g + \rho_c V_c - \rho_w (V_g + V_c) = 198 \quad \text{(2)} \]
where \( \rho_w = 1 \, \text{g/cm}^3 \) (density of water).

### Step 3: Simplify the Second Equation
Rearranging equation (2):
\[ \rho_g V_g + \rho_c V_c - \rho_w V_g - \rho_w V_c = 198 \]
Substituting the values:
\[ 19.3 V_g + 8.5 V_c - 1 (V_g + V_c) = 198 \]
This simplifies to:
\[ (19.3 - 1)V_g + (8.5 - 1)V_c = 198 \]
\[ 18.3 V_g + 7.5 V_c = 198 \quad \text{(3)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( 19.3 V_g + 8.5 V_c = 210 \) (1)
2. \( 18.3 V_g + 7.5 V_c = 198 \) (3)

We can solve these equations simultaneously. Let's solve equation (1) for \( V_c \):
\[ V_c = \frac{210 - 19.3 V_g}{8.5} \]

Substituting \( V_c \) into equation (3):
\[ 18.3 V_g + 7.5 \left(\frac{210 - 19.3 V_g}{8.5}\right) = 198 \]
Multiplying through by 8.5 to eliminate the fraction:
\[ 8.5 \times 18.3 V_g + 7.5(210 - 19.3 V_g) = 198 \times 8.5 \]
Calculating the right side:
\[ 198 \times 8.5 = 1683 \]
Now simplifying the left side:
\[ 155.55 V_g + 1575 - 144.75 V_g = 1683 \]
Combining like terms:
\[ 10.8 V_g + 1575 = 1683 \]
Subtracting 1575 from both sides:
\[ 10.8 V_g = 108 \]
Dividing by 10.8:
\[ V_g = 10 \, \text{cm}^3 \]

### Step 5: Calculate the Weight of Gold
Now we can find the weight of gold:
\[ \text{Weight of gold} = \rho_g \times V_g = 19.3 \times 10 = 193 \, \text{g} \]

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

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