How much below the surface of the earth does the acceleration due to gravity (i) reduced to `36%` (ii) reduces by `36% ` , of its value on the surface of the earth ? Radius of the earth = 6400 km .

Explanation

To solve the problem, we need to find out how deep below the surface of the Earth the acceleration due to gravity is either reduced to 36% of its surface value or reduced by 36% of its surface value.

### Given:
- Radius of the Earth, \( R = 6400 \) km
- Acceleration due to gravity at the surface, \( g \)

### Formula:
The formula for the acceleration due to gravity at a depth \( d \) below the surface of the Earth is given by:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
where \( g' \) is the acceleration due to gravity at depth \( d \).

### Case (i): Acceleration due to gravity reduced to 36% of its value on the surface

1. **Set up the equation**:
\[ g' = 0.36g \]
Substitute into the formula:
\[ 0.36g = g \left(1 - \frac{d}{R}\right) \]

2. **Cancel \( g \)** (assuming \( g \neq 0 \)):
\[ 0.36 = 1 - \frac{d}{R} \]

3. **Rearranging the equation**:
\[ \frac{d}{R} = 1 - 0.36 = 0.64 \]

4. **Solve for \( d \)**:
\[ d = 0.64R = 0.64 \times 6400 \text{ km} \]
\[ d = 4096 \text{ km} \]

### Case (ii): Acceleration due to gravity reduces by 36% of its value on the surface

1. **Set up the equation**:
\[ g' = g - 0.36g = 0.64g \]
Substitute into the formula:
\[ 0.64g = g \left(1 - \frac{d}{R}\right) \]

2. **Cancel \( g \)**:
\[ 0.64 = 1 - \frac{d}{R} \]

3. **Rearranging the equation**:
\[ \frac{d}{R} = 1 - 0.64 = 0.36 \]

4. **Solve for \( d \)**:
\[ d = 0.36R = 0.36 \times 6400 \text{ km} \]
\[ d = 2304 \text{ km} \]

### Final Answers:
- (i) The depth at which gravity reduces to 36% of its surface value is **4096 km**.
- (ii) The depth at which gravity reduces by 36% of its surface value is **2304 km**.