Calculate the percentage decrease in the weight of a body when it is taken inside a mine, 2.5 km below the earth's surface. Take,
Radius of earth = 6,400 km

Explanation

To solve the problem of calculating the percentage decrease in the weight of a body when it is taken inside a mine, 2.5 km below the Earth's surface, we can follow these steps:

### Step 1: Understand the relationship between weight and gravity
The weight \( W \) of a body is given by the formula:
\[ W = m \cdot g \]
where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity.

### Step 2: Define the percentage decrease in weight
The percentage decrease in weight can be defined as:
\[ \text{Percentage Decrease} = \frac{\Delta W}{W} \times 100 \]
where \( \Delta W \) is the change in weight, which can be expressed as:
\[ \Delta W = W_{\text{final}} - W_{\text{initial}} = m \cdot g' - m \cdot g \]
Here, \( g' \) is the acceleration due to gravity at the depth \( d \) and \( g \) is the acceleration due to gravity at the surface.

### Step 3: Cancel out mass
Since mass \( m \) is constant, we can simplify the equation:
\[ \Delta W = m \cdot (g' - g) \]
Thus, the percentage decrease in weight becomes:
\[ \text{Percentage Decrease} = \frac{m \cdot (g' - g)}{m \cdot g} \times 100 = \frac{g' - g}{g} \times 100 \]

### Step 4: Calculate the acceleration due to gravity at depth
The formula for the acceleration due to gravity at a depth \( d \) is given by:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
where \( R \) is the radius of the Earth. Given:
- \( d = 2.5 \, \text{km} = 2500 \, \text{m} \)
- \( R = 6400 \, \text{km} = 6400000 \, \text{m} \)
- \( g \approx 9.81 \, \text{m/s}^2 \)

Substituting the values:
\[ g' = 9.81 \left(1 - \frac{2500}{6400000}\right) \]

### Step 5: Calculate \( g' \)
Calculating the fraction:
\[ \frac{2500}{6400000} = 0.000390625 \]
Now substituting this back into the equation for \( g' \):
\[ g' = 9.81 \left(1 - 0.000390625\right) = 9.81 \times 0.999609375 \approx 9.807 \]

### Step 6: Calculate the percentage decrease in weight
Now we can find the percentage decrease:
\[ \text{Percentage Decrease} = \frac{g' - g}{g} \times 100 = \frac{9.807 - 9.81}{9.81} \times 100 \]
Calculating the difference:
\[ 9.807 - 9.81 = -0.003 \]
Now substituting this into the percentage decrease formula:
\[ \text{Percentage Decrease} = \frac{-0.003}{9.81} \times 100 \approx -0.0306\% \]

### Step 7: Final answer
Thus, the percentage decrease in the weight of the body when taken 2.5 km below the Earth's surface is approximately:
\[ \text{Percentage Decrease} \approx -0.0306\% \]