A man stands on vertical tower of height 20 cm. Calculate the distance up to which he will be able to see on the surface of the earth. Neglect the height of the man. Take the radius of the earth `= 6400` km

Answer

To solve the problem of how far a man standing on a vertical tower of height 20 cm can see on the surface of the Earth, we can use the formula derived from the geometry of the situation. Here’s a step-by-step solution:

### Step 1: Understand the parameters
- Height of the tower (h) = 20 cm = 0.2 m (since we need to work in meters)
- Radius of the Earth (R) = 6400 km = 6400 × 10^3 m = 6,400,000 m

### Step 2: Use the formula for distance to the horizon
The distance (d) to the horizon from a height (h) above the surface of a sphere (like the Earth) is given by the formula:
\[ d \approx \sqrt{2Rh} \]
where:
- R is the radius of the Earth
- h is the height above the surface

### Step 3: Substitute the values into the formula
Now, substituting the values of R and h into the formula:
\[ d \approx \sqrt{2 \times (6,400,000 \, \text{m}) \times (0.2 \, \text{m})} \]

### Step 4: Calculate the expression inside the square root
Calculating the product:
\[ 2 \times 6,400,000 \times 0.2 = 2,560,000 \]

### Step 5: Take the square root
Now, we take the square root of 2,560,000:
\[ d \approx \sqrt{2,560,000} \approx 1600 \, \text{m} \]

### Step 6: Convert the distance to kilometers
To convert meters to kilometers:
\[ d \approx 1600 \, \text{m} = 1.6 \, \text{km} \]

### Final Answer
The distance up to which the man will be able to see on the surface of the Earth is approximately **1.6 km**.

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