Suppose that the earth is a sphere of radius 6400 kilometers. The height from the earths surface from where exactly a fourth of the earths surface is visible, is

Options

A. `3200 km `
B. `3200 sqrt(2)` km
C. `3200 sqrt(3) ` km
D. `6400` km

Answer

To solve the problem of finding the height from which exactly a fourth of the Earth's surface is visible, we can follow these steps:

### Step 1: Understanding the Geometry
We start by visualizing the Earth as a sphere with a radius \( R = 6400 \) km. We need to find the height \( h \) above the Earth's surface from which a quarter of the Earth's surface area is visible.

### Step 2: Calculate the Total Surface Area of the Earth
The total surface area \( A \) of a sphere is given by the formula:
\[ A = 4\pi R^2 \]
Substituting the radius:
\[ A = 4\pi (6400)^2 \]

### Step 3: Determine the Visible Area
According to the problem, the area visible from height \( h \) is one-fourth of the total surface area. Therefore, the visible area \( A_{visible} \) is:
\[ A_{visible} = \frac{1}{4} A = \frac{1}{4} (4\pi R^2) = \pi R^2 \]

### Step 4: Relate the Visible Area to the Spherical Cap
The area of a spherical cap can be calculated using the formula:
\[ A_{cap} = 2\pi R h + \pi h^2 \]
However, since we are dealing with the angle subtended at the center of the sphere, we can also express the area of the spherical cap in terms of the angle \( \theta \):
\[ A_{cap} = 2\pi R^2 (1 - \cos \theta) \]
Where \( \theta \) is the angle subtended at the center of the Earth by the visible portion.

### Step 5: Set Up the Equation
From the previous steps, we have:
\[ 2\pi R^2 (1 - \cos \theta) = \pi R^2 \]
Dividing both sides by \( \pi R^2 \):
\[ 2(1 - \cos \theta) = 1 \]
This simplifies to:
\[ 1 - \cos \theta = \frac{1}{2} \]
Thus:
\[ \cos \theta = \frac{1}{2} \]

### Step 6: Find the Angle \( \theta \)
The angle \( \theta \) that satisfies \( \cos \theta = \frac{1}{2} \) is:
\[ \theta = 60^\circ \]

### Step 7: Relate the Angle to the Height
In the right triangle formed by the radius of the Earth, the height \( h \), and the line from the center of the Earth to the point of observation, we have:
\[ \cos \theta = \frac{R}{R + h} \]
Substituting \( \theta = 60^\circ \):
\[ \frac{1}{2} = \frac{R}{R + h} \]

### Step 8: Solve for \( h \)
Cross-multiplying gives:
\[ R + h = 2R \]
Thus:
\[ h = 2R - R = R \]
Since \( R = 6400 \) km, we find:
\[ h = 6400 \text{ km} \]

### Conclusion
The height from which exactly a fourth of the Earth's surface is visible is:
\[ \boxed{6400 \text{ km}} \]