A car with a horn of frequency 16 kHz is moving with a velocity of 72 km/h towards a cliff. The reflected sound heard by the driver of the car has a frequency `x xx 2000` Hz. Find the value of x if speed of sound in air is 340 m/s.
`{:(0,1,2,3,4,5,6,7,8,9),(,,,,,,,,,):}`

Explanation

To solve the problem, we will use the Doppler effect formula for sound. Here’s the step-by-step solution:

### Step 1: Convert the speed of the car from km/h to m/s
The speed of the car is given as 72 km/h. To convert this to m/s, we use the conversion factor:
\[ \text{Speed in m/s} = \text{Speed in km/h} \times \frac{5}{18} \]
Calculating:
\[ \text{Speed of the car} = 72 \times \frac{5}{18} = 20 \text{ m/s} \]

### Step 2: Identify the parameters for the Doppler effect
- Frequency of the horn (\(f_0\)) = 16 kHz = 16000 Hz
- Speed of sound in air (\(v\)) = 340 m/s
- Speed of the observer (the driver) (\(v_o\)) = 20 m/s (towards the cliff)
- Speed of the source (the car) (\(v_s\)) = 20 m/s (towards the cliff)

### Step 3: Apply the Doppler effect formula
The formula for the apparent frequency (\(f\)) when the source is moving towards a stationary observer is given by:
\[ f = f_0 \cdot \frac{v + v_o}{v - v_s} \]
In this case, the observer is moving towards the source, so we will use:
\[ f = 16000 \cdot \frac{340 + 20}{340 - 20} \]

### Step 4: Calculate the apparent frequency
Substituting the values:
\[ f = 16000 \cdot \frac{360}{320} \]
Calculating the fraction:
\[ \frac{360}{320} = 1.125 \]
Now, substituting back:
\[ f = 16000 \cdot 1.125 = 18000 \text{ Hz} \]

### Step 5: Express the frequency in terms of \(x\)
The problem states that the reflected sound heard by the driver has a frequency of \(x \times 2000\) Hz. We have found that:
\[ f = 18000 \text{ Hz} \]
Thus, we can set up the equation:
\[ 18000 = x \times 2000 \]

### Step 6: Solve for \(x\)
To find \(x\):
\[ x = \frac{18000}{2000} = 9 \]

### Final Answer
The value of \(x\) is \(9\).

---