The radius of the Earth is 6400 km. If the height of an antenna is 500 m, then its range is

Options

A. 800 km
B. 100 km
C. 80 km
D. 10 km

Answer

To solve the problem of finding the range of an antenna with a height of 500 m, given that the radius of the Earth is 6400 km, we can use the formula derived from the geometry of the situation. Here’s a step-by-step solution:

### Step 1: Understand the Geometry
The range of the antenna can be visualized as the distance from the antenna to the point where the line of sight is tangent to the Earth's surface. We can represent this scenario using a right triangle where:
- The hypotenuse is the distance from the antenna to the tangent point on the Earth's surface.
- One leg is the radius of the Earth.
- The other leg is the radius of the Earth plus the height of the antenna.

### Step 2: Define the Variables
Let:
- \( R_E \) = Radius of the Earth = 6400 km = \( 6400 \times 1000 \) m = 6,400,000 m
- \( h \) = Height of the antenna = 500 m
- \( d \) = Range of the antenna (the distance we want to find)

### Step 3: Use the Pythagorean Theorem
According to the Pythagorean theorem:
\[ (R_E + h)^2 = R_E^2 + d^2 \]

### Step 4: Expand and Rearrange the Equation
Expanding the left side:
\[ R_E^2 + 2R_Eh + h^2 = R_E^2 + d^2 \]
Subtract \( R_E^2 \) from both sides:
\[ 2R_Eh + h^2 = d^2 \]

### Step 5: Neglect the \( h^2 \) Term
Since \( h \) (500 m) is very small compared to \( R_E \) (6,400,000 m), we can neglect \( h^2 \):
\[ 2R_Eh \approx d^2 \]

### Step 6: Solve for \( d \)
Taking the square root of both sides gives:
\[ d = \sqrt{2R_Eh} \]

### Step 7: Substitute the Values
Substituting the values of \( R_E \) and \( h \):
\[ d = \sqrt{2 \times 6,400,000 \, \text{m} \times 500 \, \text{m}} \]

### Step 8: Calculate the Range
Calculating the expression:
\[ d = \sqrt{2 \times 6,400,000 \times 500} = \sqrt{6,400,000,0000} = \sqrt{6.4 \times 10^{12}} = 80,000 \, \text{m} \]
Converting meters to kilometers:
\[ d = 80 \, \text{km} \]

### Conclusion
Thus, the range of the antenna is **80 km**.

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