The radius of the earth is 6,400 km. Calculate the height from the surface of the earth at which the value of g is 81% of the value at the surface.

Explanation

To solve the problem, we need to find the height \( h \) from the surface of the Earth at which the value of \( g \) is 81% of its value at the surface.

### Step 1: Understand the relationship between \( g \) at height \( h \) and at the surface
The acceleration due to gravity at a height \( h \) above the Earth's surface is given by the formula:

\[ g' = \frac{g}{(1 + \frac{h}{R})^2} \]

where:
- \( g' \) is the acceleration due to gravity at height \( h \),
- \( g \) is the acceleration due to gravity at the surface of the Earth,
- \( R \) is the radius of the Earth.

### Step 2: Set up the equation
We know that \( g' \) is 81% of \( g \):

\[ g' = 0.81g \]

Substituting this into the equation gives:

\[ 0.81g = \frac{g}{(1 + \frac{h}{R})^2} \]

### Step 3: Cancel \( g \) from both sides
Assuming \( g \neq 0 \), we can divide both sides by \( g \):

\[ 0.81 = \frac{1}{(1 + \frac{h}{R})^2} \]

### Step 4: Rearrange the equation
Taking the reciprocal of both sides:

\[ (1 + \frac{h}{R})^2 = \frac{1}{0.81} \]

### Step 5: Calculate \( \frac{1}{0.81} \)
Calculating \( \frac{1}{0.81} \):

\[ \frac{1}{0.81} \approx 1.2346 \]

### Step 6: Take the square root
Now, take the square root of both sides:

\[ 1 + \frac{h}{R} = \sqrt{1.2346} \]

Calculating \( \sqrt{1.2346} \):

\[ \sqrt{1.2346} \approx 1.11 \]

### Step 7: Solve for \( h \)
Now, we can solve for \( h \):

\[ \frac{h}{R} = 1.11 - 1 \]

\[ \frac{h}{R} = 0.11 \]

Multiplying both sides by \( R \):

\[ h = 0.11R \]

### Step 8: Substitute the radius of the Earth
Given that the radius of the Earth \( R = 6400 \) km:

\[ h = 0.11 \times 6400 \text{ km} \]

Calculating \( h \):

\[ h \approx 704 \text{ km} \]

### Final Answer
The height from the surface of the Earth at which the value of \( g \) is 81% of the value at the surface is approximately **704 km**.

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