The approximate height from the surface of earth at which the weight of the body becomes `1/3` of its weight on the surface of earth is :
[Radius of earth R=6400 km and `sqrt3 = 1.732` ]

Options

A. 3840 km
B. 4685 km
C. 2133 km
D. 4267 km

Answer

To find the approximate height from the surface of the Earth at which the weight of a body becomes \( \frac{1}{3} \) of its weight on the surface of the Earth, we can follow these steps:

### Step 1: Understanding the Problem
The weight of a body on the surface of the Earth is given by the gravitational force acting on it, which is proportional to the acceleration due to gravity \( g \). At a height \( h \) above the Earth's surface, the acceleration due to gravity \( g_h \) can be expressed as:
\[ g_h = \frac{g}{(1 + \frac{h}{R})^2} \]
where \( R \) is the radius of the Earth.

### Step 2: Setting Up the Equation
We want to find the height \( h \) where the weight becomes \( \frac{1}{3} \) of its weight on the surface. This means:
\[ g_h = \frac{g}{3} \]
Substituting this into our equation gives:
\[ \frac{g}{(1 + \frac{h}{R})^2} = \frac{g}{3} \]

### Step 3: Simplifying the Equation
We can cancel \( g \) from both sides (assuming \( g \neq 0 \)):
\[ \frac{1}{(1 + \frac{h}{R})^2} = \frac{1}{3} \]
Taking the reciprocal of both sides results in:
\[ (1 + \frac{h}{R})^2 = 3 \]

### Step 4: Taking the Square Root
Taking the square root of both sides gives:
\[ 1 + \frac{h}{R} = \sqrt{3} \]

### Step 5: Solving for \( h \)
Now, we can isolate \( h \):
\[ \frac{h}{R} = \sqrt{3} - 1 \]
\[ h = R(\sqrt{3} - 1) \]

### Step 6: Substituting the Value of \( R \)
Given that the radius of the Earth \( R = 6400 \) km, we substitute this value in:
\[ h = 6400(\sqrt{3} - 1) \]
Using the approximation \( \sqrt{3} \approx 1.732 \):
\[ h = 6400(1.732 - 1) = 6400(0.732) \]

### Step 7: Calculating \( h \)
Now, we calculate:
\[ h \approx 6400 \times 0.732 \approx 4686.4 \text{ km} \]

### Conclusion
Thus, the approximate height from the surface of the Earth at which the weight of the body becomes \( \frac{1}{3} \) of its weight on the surface of the Earth is approximately:
\[ \boxed{4686 \text{ km}} \]