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 .

Answer

To solve the problem, we need to determine the depth below the Earth's surface at which 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, \( g \)

### Part (i): When the acceleration due to gravity is reduced to 36% of its value on the surface of the Earth.

1. **Understanding the relationship**:
The acceleration due to gravity at a depth \( d \) below the surface 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 \( d \).

2. **Setting up the equation**:
We want to find the depth \( d \) where \( g_d = 0.36g \):
\[ 0.36g = g \left(1 - \frac{d}{R}\right) \]

3. **Cancelling \( g \)**:
Since \( g \) is common on both sides, we can cancel it out:
\[ 0.36 = 1 - \frac{d}{R} \]

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

5. **Finding the depth \( d \)**:
Now, multiply both sides by \( R \):
\[ d = 0.64R = 0.64 \times 6400 \text{ km} = 4096 \text{ km} \]

### Part (ii): When the acceleration due to gravity is reduced by 36% of its value on the surface of the Earth.

1. **Understanding the relationship**:
In this case, the acceleration due to gravity is reduced by 36%, meaning it retains 64% of its original value:
\[ g_d = 0.64g \]

2. **Setting up the equation**:
Using the same formula:
\[ 0.64g = g \left(1 - \frac{d'}{R}\right) \]

3. **Cancelling \( g \)**:
Again, we can cancel \( g \):
\[ 0.64 = 1 - \frac{d'}{R} \]

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

5. **Finding the depth \( d' \)**:
Now, multiply both sides by \( R \):
\[ d' = 0.36R = 0.36 \times 6400 \text{ km} = 2304 \text{ km} \]

### Final Answers:
- (i) The depth where \( g \) is reduced to 36% of its surface value is **4096 km**.
- (ii) The depth where \( g \) is reduced by 36% of its surface value is **2304 km**.