Sure, let's solve this problem step-by-step using the Doppler effect formula.
### Step 1: Understand the Doppler effect formula
The Doppler effect is an observed change in frequency when the source of the sound is moving relative to the observer. The formula for the observed frequency [tex]\( f \)[/tex] when a source is moving towards a stationary observer is:
[tex]\[ f = f_0 \times \frac{v_{sound}}{v_{sound} - v_{source}} \][/tex]
where:
- [tex]\( f_0 \)[/tex] is the emitted (original) frequency of the sound,
- [tex]\( v_{sound} \)[/tex] is the speed of sound in the medium,
- [tex]\( v_{source} \)[/tex] is the speed of the source moving towards the observer.
### Step 2: Set up the equations for the two cases
For the first case where the train is moving towards the observer with a speed of [tex]\( 34 \, \text{m/s} \)[/tex]:
[tex]\[ f_1 = f_0 \times \frac{v_{sound}}{v_{sound} - 34} \][/tex]
For the second case where the train slows down to a speed of [tex]\( 17 \, \text{m/s} \)[/tex]:
[tex]\[ f_2 = f_0 \times \frac{v_{sound}}{v_{sound} - 17} \][/tex]
### Step 3: Find the ratio [tex]\( \frac{f_1}{f_2} \)[/tex]
To find the ratio [tex]\( \frac{f_1}{f_2} \)[/tex], we divide the two equations:
[tex]\[ \frac{f_1}{f_2} = \frac{f_0 \times \frac{v_{sound}}{v_{sound} - 34}}{f_0 \times \frac{v_{sound}}{v_{sound} - 17}} \][/tex]
Simplifying, we can cancel out [tex]\( f_0 \)[/tex] and [tex]\( v_{sound} \)[/tex]:
[tex]\[ \frac{f_1}{f_2} = \frac{\frac{v_{sound}}{v_{sound} - 34}}{\frac{v_{sound}}{v_{sound} - 17}} \][/tex]
[tex]\[ \frac{f_1}{f_2} = \frac{v_{sound}}{v_{sound} - 34} \cdot \frac{v_{sound} - 17}{v_{sound}} \][/tex]
[tex]\[ \frac{f_1}{f_2} = \frac{v_{sound} - 17}{v_{sound} - 34} \][/tex]
Given:
[tex]\[ v_{sound} = 340 \, \text{m/s} \][/tex]
Substitute this value:
[tex]\[ \frac{f_1}{f_2} = \frac{340 - 17}{340 - 34} \][/tex]
[tex]\[ \frac{f_1}{f_2} = \frac{323}{306} \][/tex]
### Step 4: Simplify the ratio
[tex]\[ 323 \div 17 = 19 \][/tex]
[tex]\[ 306 \div 17 = 18 \][/tex]
So, the simplified ratio is:
[tex]\[ \frac{323}{306} = \frac{19}{18} \][/tex]
Therefore, the ratio [tex]\( \frac{f_1}{f_2} \)[/tex] is [tex]\( \frac{19}{18} \)[/tex], which corresponds to option (d).
### Final Answer:
The ratio [tex]\( \frac{f_1}{f_2} \)[/tex] is [tex]\( \frac{19}{18} \)[/tex].
