Find the approximate ratio of frequencies of a microwave and an ultrasonic sound wave that have the same wavelength.

Options

A. `10^(2)`
B. `10^(4)`
C. `10^(6)`
D. `10^(8)`

Explanation

To find the approximate ratio of frequencies of a microwave and an ultrasonic sound wave that have the same wavelength, we can follow these steps:

### Step 1: Understand the relationship between wavelength and frequency
The relationship between the speed of a wave (v), its wavelength (λ), and its frequency (f) is given by the equation:
\[ v = f \cdot \lambda \]
Where:
- \( v \) is the speed of the wave,
- \( f \) is the frequency,
- \( \lambda \) is the wavelength.

### Step 2: Identify the speeds of microwaves and ultrasonic sound waves
- The speed of microwaves in a vacuum (or air) is approximately \( 3 \times 10^8 \) m/s (the speed of light).
- The speed of sound in air (for ultrasonic waves) is approximately \( 3.4 \times 10^2 \) m/s.

### Step 3: Set up the equations for frequency
Since both waves have the same wavelength (let's denote it as \( \lambda \)), we can express the frequencies of both waves using the speed and wavelength:
1. For microwaves:
\[ f_{microwave} = \frac{v_{microwave}}{\lambda} = \frac{3 \times 10^8 \, \text{m/s}}{\lambda} \]
2. For ultrasonic waves:
\[ f_{ultrasonic} = \frac{v_{ultrasonic}}{\lambda} = \frac{3.4 \times 10^2 \, \text{m/s}}{\lambda} \]

### Step 4: Find the ratio of frequencies
Now, we can find the ratio of the frequencies:
\[ \frac{f_{microwave}}{f_{ultrasonic}} = \frac{\frac{3 \times 10^8}{\lambda}}{\frac{3.4 \times 10^2}{\lambda}} = \frac{3 \times 10^8}{3.4 \times 10^2} \]

### Step 5: Simplify the ratio
Now, we can simplify this expression:
\[ \frac{f_{microwave}}{f_{ultrasonic}} = \frac{3 \times 10^8}{3.4 \times 10^2} = \frac{3}{3.4} \times 10^{8-2} = \frac{3}{3.4} \times 10^6 \]
Calculating \( \frac{3}{3.4} \) gives approximately \( 0.882 \). Therefore:
\[ \frac{f_{microwave}}{f_{ultrasonic}} \approx 0.882 \times 10^6 \approx 8.82 \times 10^5 \]

### Step 6: Conclusion
Thus, the approximate ratio of the frequencies of a microwave to an ultrasonic sound wave that have the same wavelength is:
\[ \frac{f_{microwave}}{f_{ultrasonic}} \approx 10^6 \]