To cover a population of 20 lakh, a transmission tower should have a height of (radius of the earth = 6,400 km, population `1000//km^(2)`).

Options

A. 25 m
B. 50 m
C. 75 m
D. 100 m

Answer

To solve the problem of determining the height of a transmission tower required to cover a population of 20 lakh, we can follow these steps:

### Step 1: Understand the Population and Density
- Given that the population is 20 lakh (which is 20 x 10^5 = 2,000,000 people).
- The population density is given as 1000 people per square kilometer.

### Step 2: Calculate the Area Required
- To find the area required to accommodate 20 lakh people, we can use the formula:

\[ \text{Area} = \frac{\text{Population}}{\text{Population Density}} = \frac{2,000,000 \text{ people}}{1000 \text{ people/km}^2} = 2000 \text{ km}^2 \]

### Step 3: Relate Area to the Radius
- The area of a circle is given by the formula:

\[ \text{Area} = \pi r^2 \]

- We can set this equal to the area we calculated:

\[ \pi r^2 = 2000 \text{ km}^2 \]

### Step 4: Solve for the Radius
- Rearranging the equation to solve for \( r \):

\[ r^2 = \frac{2000}{\pi} \]

- Taking the square root:

\[ r = \sqrt{\frac{2000}{\pi}} \text{ km} \]

### Step 5: Use the Relationship Between Height and Radius
- The relationship between the height of the tower (h) and the radius (r) is given by the formula:

\[ d = \sqrt{2rh} \]

- Here, \( d \) is the diameter (which is \( 2r \)), so we can rewrite the equation as:

\[ 2r = \sqrt{2rh} \]

### Step 6: Substitute and Solve for Height
- Squaring both sides gives:

\[ (2r)^2 = 2rh \]

- Simplifying this gives:

\[ 4r^2 = 2rh \]

- Rearranging to solve for \( h \):

\[ h = \frac{4r^2}{2r} = 2r \]

### Step 7: Substitute the Value of r
- Now substituting \( r = \sqrt{\frac{2000}{\pi}} \):

\[ h = 2 \cdot \sqrt{\frac{2000}{\pi}} \]

### Step 8: Calculate the Height
- Using the value of \( \pi \approx 3.14 \):

\[ h = 2 \cdot \sqrt{\frac{2000}{3.14}} \approx 2 \cdot \sqrt{636.94} \approx 2 \cdot 25.24 \approx 50.48 \text{ meters} \]

- Rounding this gives approximately 50 meters.

### Final Answer
Thus, the height of the transmission tower should be approximately **50 meters**.

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