A sphere is made of an alloy of Metal A (density `8 g//cm^(3)`) and Metal B (density `6g//cm^(3)`). The sphere floats in mercury (density `13.6 g//cm^(3)`) with half its volume submerged. The percentage of the total volume of the sphere that is occupied by metal A is ___________ .

Explanation

To solve the problem, we need to find the percentage of the total volume of the sphere that is occupied by metal A. We will follow these steps:

### Step 1: Understand the buoyancy condition
Since the sphere floats in mercury with half of its volume submerged, we can use the principle of buoyancy. The buoyant force acting on the sphere is equal to the weight of the sphere.

### Step 2: Set up the equations
Let the total volume of the sphere be \( V \). The weight of the sphere can be expressed as:
\[ \text{Weight of sphere} = \rho_{\text{alloy}} \cdot V \cdot g \]
where \( \rho_{\text{alloy}} \) is the density of the alloy and \( g \) is the acceleration due to gravity.

The buoyant force acting on the sphere is given by:
\[ \text{Buoyant force} = \rho_{\text{mercury}} \cdot \left(\frac{V}{2}\right) \cdot g \]
where \( \rho_{\text{mercury}} = 13.6 \, \text{g/cm}^3 \).

### Step 3: Equate the weight and buoyant force
Setting the weight of the sphere equal to the buoyant force, we have:
\[ \rho_{\text{alloy}} \cdot V \cdot g = \rho_{\text{mercury}} \cdot \left(\frac{V}{2}\right) \cdot g \]
We can cancel \( V \) and \( g \) from both sides:
\[ \rho_{\text{alloy}} = \frac{\rho_{\text{mercury}}}{2} \]

### Step 4: Calculate the density of the alloy
Substituting the value of \( \rho_{\text{mercury}} \):
\[ \rho_{\text{alloy}} = \frac{13.6 \, \text{g/cm}^3}{2} = 6.8 \, \text{g/cm}^3 \]

### Step 5: Set up the equation for the alloy's density
Let the fraction of the volume occupied by metal A be \( x \). Therefore, the fraction occupied by metal B will be \( 1 - x \). The densities of metals A and B are given as:
- \( \rho_A = 8 \, \text{g/cm}^3 \)
- \( \rho_B = 6 \, \text{g/cm}^3 \)

The density of the alloy can be expressed as:
\[ \rho_{\text{alloy}} = \frac{\rho_A \cdot x + \rho_B \cdot (1 - x)}{1} \]
Substituting the known values:
\[ 6.8 = 8x + 6(1 - x) \]

### Step 6: Simplify and solve for \( x \)
Expanding the equation:
\[ 6.8 = 8x + 6 - 6x \]
\[ 6.8 = 2x + 6 \]
Subtracting 6 from both sides:
\[ 0.8 = 2x \]
Dividing by 2:
\[ x = 0.4 \]

### Step 7: Calculate the percentage of volume occupied by metal A
To find the percentage of the total volume of the sphere that is occupied by metal A:
\[ \text{Percentage of A} = x \cdot 100 = 0.4 \cdot 100 = 40\% \]

### Final Answer
The percentage of the total volume of the sphere that is occupied by metal A is **40%**.