A remote sensing satellite is revolving in an orbit of radius x on the equator of earth. Find the area on earth surface in which satellite can not send messsage.

Explanation

To solve the problem of finding the area on the Earth's surface where a remote sensing satellite cannot send messages, we can follow these steps:

### Step 1: Understand the Geometry of the Problem
The satellite is in orbit at a radius \( x \) from the center of the Earth, which has a radius \( R \). The satellite's position allows it to communicate with a certain area on the Earth's surface, but there are regions (specifically the polar regions) where communication is not possible.

### Step 2: Determine the Angle of Communication
The satellite can communicate with the surface of the Earth up to a certain angle \( \theta \) from the vertical. This angle can be determined using the geometry of the situation. The maximum reach of the satellite is determined by the line of sight from the satellite to the Earth's surface.

### Step 3: Use Trigonometry to Find \( \cos \theta \)
From the geometry, we can use the cosine of the angle \( \theta \):
\[ \cos \theta = \frac{R}{x} \]
This implies that:
\[ \theta = \cos^{-1}\left(\frac{R}{x}\right) \]

### Step 4: Calculate the Area That Cannot Be Covered
The area on the Earth's surface that cannot be reached by the satellite includes the polar regions and the area subtended by the angle \( \theta \). The area of the spherical cap that cannot be reached can be calculated using the formula:
\[ \text{Area}_{\text{not covered}} = 2\pi R^2 (1 - \cos \theta) \]
Substituting \( \cos \theta \):
\[ \text{Area}_{\text{not covered}} = 2\pi R^2 \left(1 - \frac{R}{x}\right) \]

### Step 5: Final Area Calculation
The total area of the Earth is \( 4\pi R^2 \). Thus, the area that the satellite can cover is:
\[ \text{Area}_{\text{covered}} = 4\pi R^2 - \text{Area}_{\text{not covered}} \]
However, since we are interested in the area that cannot be covered:
\[ \text{Area}_{\text{not covered}} = 2\pi R^2 \left(1 - \frac{R}{x}\right) \]

### Final Answer
The area on the Earth's surface where the satellite cannot send messages is:
\[ \text{Area}_{\text{not covered}} = 2\pi R^2 \left(1 - \frac{R}{x}\right) \]