If a man weighs 90 kg on the surface of the earth, the height above the surface of the earth of radius, R where the weight is 30 kg is

Options

A. 0.73 R
B. `sqrt(3) R`
C. `R`
D. `2R`

Answer

To solve the problem, we need to find the height above the Earth's surface where a man weighing 90 kg on the surface weighs only 30 kg.

### Step-by-Step Solution:

1. **Understanding Weight**:
The weight of an object is given by the formula:
\[ W = m \cdot g \]
where \( W \) is the weight, \( m \) is the mass, and \( g \) is the acceleration due to gravity.

2. **Weight on Earth's Surface**:
On the surface of the Earth, the man weighs 90 kg. Therefore, we can express this as:
\[ W_{\text{surface}} = m \cdot g_{\text{surface}} = 90 \, \text{kg} \]

3. **Weight at Height \( h \)**:
At a height \( h \) above the Earth's surface, the weight of the man is given as 30 kg. Thus:
\[ W_{\text{height}} = m \cdot g_{\text{height}} = 30 \, \text{kg} \]

4. **Relating Gravity at Height**:
The acceleration due to gravity at a height \( h \) is given by the formula:
\[ g_{\text{height}} = \frac{g_{\text{surface}} \cdot R^2}{(R + h)^2} \]
where \( R \) is the radius of the Earth.

5. **Setting Up the Equation**:
From the weight equations, we can set up the following relationship:
\[ 30 = 90 \cdot \frac{g_{\text{surface}} \cdot R^2}{(R + h)^2} \]
Simplifying this gives:
\[ \frac{1}{3} = \frac{R^2}{(R + h)^2} \]

6. **Cross-Multiplying**:
Cross-multiplying gives:
\[ (R + h)^2 = 3R^2 \]

7. **Taking the Square Root**:
Taking the square root of both sides results in:
\[ R + h = \sqrt{3} R \]

8. **Solving for \( h \)**:
Rearranging gives:
\[ h = \sqrt{3} R - R = (\sqrt{3} - 1) R \]

9. **Calculating the Height**:
To find the numerical value of \( h \), we can approximate \( \sqrt{3} \) as 1.732:
\[ h \approx (1.732 - 1) R = 0.732 R \]

### Final Answer:
Thus, the height \( h \) above the surface of the Earth where the weight of the man is 30 kg is:
\[ h \approx 0.732 R \]