The depth from the surface of the earth where acceleration due to gravity is `20%` of its value on the surface of the earth is (R = 6400 km)

Options

A. 1280 km
B. 5120 km
C. 800 km
D. 640 km

Explanation

To solve the problem of finding the depth from the surface of the Earth where the acceleration due to gravity is 20% of its value on the surface, we can follow these steps:

### Step 1: Understand the relationship between gravity at depth and surface
The acceleration due to gravity at a depth \( d \) below the surface of the Earth is given by the formula:
\[ g_d = g \left(1 - \frac{d}{R}\right) \]
where:
- \( g_d \) is the acceleration due to gravity at depth \( d \),
- \( g \) is the acceleration due to gravity at the surface (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( R \) is the radius of the Earth (given as \( 6400 \, \text{km} \)).

### Step 2: Set up the equation for 20% of surface gravity
We are looking for the depth where the gravity is 20% of the surface gravity:
\[ g_d = 0.2g \]
Substituting the expression for \( g_d \):
\[ g \left(1 - \frac{d}{R}\right) = 0.2g \]

### Step 3: Simplify the equation
We can cancel \( g \) from both sides (assuming \( g \neq 0 \)):
\[ 1 - \frac{d}{R} = 0.2 \]

### Step 4: Solve for \( d \)
Rearranging the equation gives:
\[ \frac{d}{R} = 1 - 0.2 = 0.8 \]
Thus, we can express \( d \) as:
\[ d = 0.8R \]

### Step 5: Substitute the value of \( R \)
Now, substituting the value of \( R = 6400 \, \text{km} \):
\[ d = 0.8 \times 6400 \, \text{km} = 5120 \, \text{km} \]

### Conclusion
The depth from the surface of the Earth where the acceleration due to gravity is 20% of its value on the surface is:
\[ \boxed{5120 \, \text{km}} \]