A solid shell loses half of its weight in water. Relative density of shell is 5.0 What fraction of its volume is hollow ?

Options

A. `(3)/(5)`
B. `(2)/(5)`
C. `(1)/(5)`
D. `(4)/(5)`

Answer

To solve the problem, we need to determine the fraction of the volume of the solid shell that is hollow, given that the shell loses half of its weight in water and has a relative density of 5.0.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The solid shell loses half of its weight when submerged in water.
- The relative density (specific gravity) of the shell is 5.0, which means the density of the shell is 5 times the density of water.

2. **Define Variables**:
- Let \( V \) be the total volume of the shell.
- Let \( V_c \) be the volume of the hollow part of the shell.
- The volume of the solid part of the shell will then be \( V - V_c \).

3. **Weight of the Shell**:
- The weight of the shell in air is given by:
\[ W = \text{Density of shell} \times V \times g = (5 \times 10^3) \times V \times g \]
- This is because the density of water is \( 10^3 \, \text{kg/m}^3 \).

4. **Loss of Weight in Water**:
- When submerged in water, the shell loses half of its weight:
\[ \text{Loss of weight} = \frac{W}{2} = \frac{(5 \times 10^3) V g}{2} \]

5. **Buoyant Force**:
- The buoyant force acting on the shell when it is submerged is equal to the weight of the water displaced:
\[ \text{Buoyant Force} = \text{Density of water} \times \text{Volume displaced} \times g = (10^3) \times V \times g \]

6. **Setting Up the Equation**:
- Since the loss of weight is equal to the buoyant force, we can set up the equation:
\[ \frac{(5 \times 10^3) V g}{2} = (10^3) \times V \times g \]
- Cancel \( g \) from both sides:
\[ \frac{(5 \times 10^3) V}{2} = (10^3) V \]

7. **Simplifying the Equation**:
- Multiply both sides by 2 to eliminate the fraction:
\[ 5 \times 10^3 V = 2 \times 10^3 V \]
- Rearranging gives:
\[ 5V - 2V = 0 \implies 3V = 5V_c \]

8. **Finding the Fraction of Hollow Volume**:
- From the equation \( 3V = 5V_c \), we can express \( V_c \) in terms of \( V \):
\[ V_c = \frac{3V}{5} \]
- The fraction of the volume that is hollow is:
\[ \frac{V_c}{V} = \frac{3V/5}{V} = \frac{3}{5} \]

### Final Answer:
The fraction of the volume that is hollow is \( \frac{3}{5} \).