Calculate the percentage decrease in the weight of a body when it is taken to a height of 40 km above the earth's surface. Radius of earth = 6,400 km.

Explanation

To calculate the percentage decrease in the weight of a body when it is taken to a height of 40 km above the Earth's surface, we can follow these steps:

### Step 1: Understand the relationship between weight and distance from the center of the Earth.
The weight of an object is given by the formula:
\[ W = \frac{G \cdot m \cdot M}{r^2} \]
where:
- \( W \) is the weight,
- \( G \) is the gravitational constant,
- \( m \) is the mass of the object,
- \( M \) is the mass of the Earth,
- \( r \) is the distance from the center of the Earth.

### Step 2: Calculate the initial weight at the Earth's surface.
At the surface of the Earth, the distance \( r \) is equal to the radius of the Earth, which is 6400 km. Thus, the weight at the surface \( W_1 \) can be expressed as:
\[ W_1 = \frac{G \cdot m \cdot M}{(6400 \text{ km})^2} \]

### Step 3: Calculate the new distance from the center of the Earth at a height of 40 km.
When the body is taken to a height of 40 km, the new distance \( r \) from the center of the Earth becomes:
\[ r = 6400 \text{ km} + 40 \text{ km} = 6440 \text{ km} \]

### Step 4: Calculate the weight at the new height.
The weight at the height of 40 km \( W_2 \) can be expressed as:
\[ W_2 = \frac{G \cdot m \cdot M}{(6440 \text{ km})^2} \]

### Step 5: Calculate the percentage decrease in weight.
The percentage decrease in weight can be calculated using the formula:
\[ \text{Percentage Decrease} = \frac{W_1 - W_2}{W_1} \times 100\% \]

### Step 6: Substitute the values to find \( W_1 \) and \( W_2 \).
1. Calculate \( W_1 \):
\[ W_1 = \frac{G \cdot m \cdot M}{(6400)^2} \]

2. Calculate \( W_2 \):
\[ W_2 = \frac{G \cdot m \cdot M}{(6440)^2} \]

### Step 7: Plug these into the percentage decrease formula.
\[ \text{Percentage Decrease} = \frac{\frac{G \cdot m \cdot M}{(6400)^2} - \frac{G \cdot m \cdot M}{(6440)^2}}{\frac{G \cdot m \cdot M}{(6400)^2}} \times 100\% \]

### Step 8: Simplify the expression.
This simplifies to:
\[ \text{Percentage Decrease} = \left(1 - \frac{(6400)^2}{(6440)^2}\right) \times 100\% \]

### Step 9: Calculate the numerical values.
1. Calculate \( (6400)^2 \) and \( (6440)^2 \):
- \( (6400)^2 = 40960000 \)
- \( (6440)^2 = 41473600 \)

2. Substitute these values:
\[ \text{Percentage Decrease} = \left(1 - \frac{40960000}{41473600}\right) \times 100\% \]

3. Calculate the fraction:
\[ \frac{40960000}{41473600} \approx 0.9875 \]

4. Calculate the percentage:
\[ \text{Percentage Decrease} = (1 - 0.9875) \times 100\% \approx 1.25\% \]

### Final Answer:
The percentage decrease in the weight of the body when taken to a height of 40 km above the Earth's surface is approximately **1.25%**.

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