A microwave telephone link operating at the central frequency of 10GHz has been established if 2% of this is available for microwave communication channel. Then how many telephone channels can be simultaneously granted if each telephone is allotted a bandwidth of 8 KHz?

Options

A. `1.5xx10^(3)`
B. `3.5xx10^(2)`
C. `2.5xx10^(4)`
D. `4.5xx10^(6)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Determine the total frequency available for communication.
The central frequency given is 10 GHz. We need to convert this into hertz:
\[ 10 \text{ GHz} = 10 \times 10^9 \text{ Hz} = 10^{10} \text{ Hz} \]

### Step 2: Calculate the available frequency for the communication channel.
We are told that 2% of this frequency is available for microwave communication. Thus, we calculate:
\[ \text{Available frequency} = 2\% \text{ of } 10^{10} \text{ Hz} = \frac{2}{100} \times 10^{10} \text{ Hz} = 0.02 \times 10^{10} \text{ Hz} = 2 \times 10^8 \text{ Hz} \]

### Step 3: Determine the bandwidth allotted to each telephone channel.
Each telephone is allotted a bandwidth of 8 kHz. We need to convert this into hertz:
\[ 8 \text{ kHz} = 8 \times 10^3 \text{ Hz} \]

### Step 4: Calculate the number of telephone channels that can be simultaneously granted.
To find the number of channels, we divide the available frequency by the bandwidth allotted to each channel:
\[ \text{Number of channels} = \frac{\text{Available frequency}}{\text{Allotted bandwidth}} = \frac{2 \times 10^8 \text{ Hz}}{8 \times 10^3 \text{ Hz}} \]

### Step 5: Simplify the calculation.
\[ \text{Number of channels} = \frac{2 \times 10^8}{8 \times 10^3} = \frac{2}{8} \times \frac{10^8}{10^3} = 0.25 \times 10^{5} = 2.5 \times 10^{4} \]

### Conclusion:
The number of telephone channels that can be simultaneously granted is:
\[ \text{Number of channels} = 2.5 \times 10^4 \]

### Final Answer:
Thus, the answer is \( 25000 \) telephone channels.

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