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 need to determine how many telephone channels can be simultaneously granted given the available bandwidth for communication. Here’s a step-by-step solution:

### Step 1: Determine the central frequency
The central frequency of the microwave telephone link is given as:
\[ \nu = 10 \text{ GHz} \]
Converting this to Hertz:
\[ \nu = 10 \times 10^9 \text{ Hz} = 10^{10} \text{ Hz} \]

**Hint:** Remember that 1 GHz = \(10^9\) Hz.

### Step 2: Calculate the available bandwidth
We know that only 2% of this frequency is available for the microwave communication channel. Therefore, we calculate 2% of \(10^{10}\) Hz:
\[ \text{Available Bandwidth} = \frac{2}{100} \times 10^{10} \text{ Hz} \]
\[ = 0.02 \times 10^{10} \text{ Hz} = 2 \times 10^8 \text{ Hz} \]

**Hint:** To find a percentage, multiply the total by the percentage in decimal form.

### Step 3: Determine the bandwidth allotted per channel
Each telephone channel is allotted a bandwidth of:
\[ \text{Bandwidth per channel} = 8 \text{ kHz} \]
Converting this to Hertz:
\[ \text{Bandwidth per channel} = 8 \times 10^3 \text{ Hz} \]

**Hint:** Remember that 1 kHz = \(10^3\) Hz.

### Step 4: Calculate the number of channels
Now, we can find the number of telephone channels \(n\) that can be simultaneously granted using the formula:
\[ n = \frac{\text{Total Bandwidth}}{\text{Bandwidth per channel}} \]
Substituting the values we calculated:
\[ n = \frac{2 \times 10^8 \text{ Hz}}{8 \times 10^3 \text{ Hz}} \]

### Step 5: Simplify the calculation
Now, we simplify the expression:
\[ n = \frac{2 \times 10^8}{8 \times 10^3} = \frac{2}{8} \times \frac{10^8}{10^3} = \frac{1}{4} \times 10^{5} = 0.25 \times 10^{5} = 2.5 \times 10^{4} \]

**Hint:** When dividing powers of ten, subtract the exponents.

### Step 6: Convert to a more understandable number
Finally, we can express \(2.5 \times 10^{4}\) in standard form:
\[ n = 25000 \]

Thus, the number of telephone channels that can be simultaneously granted is **25000**.

**Final Answer:** 25000 telephone channels.

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