A metallic sphere with an internal cavity weighs `40 gwt` in air and `20 gwt` in water. If the density of the material with cavity be `8g per cm^(3)` then the volume of cavity is :

Options

A. Zero
B. `15 cm^(3)`
C. `5 cm^(3)`
D. `20 cm^(3)`

Explanation

To find the volume of the cavity in the metallic sphere, we can follow these steps:

### Step 1: Understand the Problem
The metallic sphere weighs 40 gwt in air and 20 gwt in water. We need to find the volume of the internal cavity of the sphere. We are also given the density of the material of the sphere as 8 g/cm³.

### Step 2: Calculate the Weight of Liquid Displaced
The apparent weight of the sphere in water is given as 20 gwt. The true weight in air is 40 gwt. The weight of the liquid displaced can be calculated using the formula:
\[ \text{Weight of liquid displaced} = \text{True weight} - \text{Apparent weight} \]
\[ \text{Weight of liquid displaced} = 40 \, \text{gwt} - 20 \, \text{gwt} = 20 \, \text{gwt} \]

### Step 3: Calculate the Volume of Water Displaced
Since the density of water is 1 g/cm³, the volume of water displaced can be calculated using the formula:
\[ \text{Volume of water displaced} = \frac{\text{Weight of liquid displaced}}{\text{Density of water}} \]
\[ \text{Volume of water displaced} = \frac{20 \, \text{g}}{1 \, \text{g/cm}^3} = 20 \, \text{cm}^3 \]

### Step 4: Calculate the Volume of the Material of the Sphere
The true weight of the sphere is 40 gwt, and the density of the material is 8 g/cm³. The volume of the material can be calculated as:
\[ \text{Volume of material} = \frac{\text{True weight}}{\text{Density of material}} \]
\[ \text{Volume of material} = \frac{40 \, \text{g}}{8 \, \text{g/cm}^3} = 5 \, \text{cm}^3 \]

### Step 5: Calculate the Volume of the Cavity
Now, we can find the volume of the cavity using the volumes calculated:
\[ \text{Volume of cavity} = \text{Volume of water displaced} - \text{Volume of material} \]
\[ \text{Volume of cavity} = 20 \, \text{cm}^3 - 5 \, \text{cm}^3 = 15 \, \text{cm}^3 \]

### Final Answer
The volume of the cavity in the metallic sphere is **15 cm³**.

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