Certainly! Let's delve into the steps required to find the length of the shortest closed pipe that would resonate at 218 Hz, given the speed of sound as 340 m/s.
### Step-by-Step Solution:
1. Given Data:
- Speed of sound ([tex]\( v \)[/tex]) = 340 m/s
- Frequency ([tex]\( f \)[/tex]) = 218 Hz
2. Identify the formula:
- For a closed pipe, the fundamental frequency (1st harmonic) can be calculated using:
[tex]\[
f = \frac{v}{4L}
\][/tex]
where:
- [tex]\( f \)[/tex] is the frequency
- [tex]\( v \)[/tex] is the speed of sound
- [tex]\( L \)[/tex] is the length of the pipe
3. Rearrange the formula to solve for [tex]\( L \)[/tex]:
[tex]\[
L = \frac{v}{4f}
\][/tex]
4. Substitute the given values into the equation:
[tex]\[
L = \frac{340 \, \text{m/s}}{4 \times 218 \, \text{Hz}}
\][/tex]
5. Calculate the length [tex]\( L \)[/tex]:
[tex]\[
L = \frac{340}{872}
\][/tex]
[tex]\[
L = 0.38990825688073394 \, \text{meters}
\][/tex]
6. Convert the length from meters to centimeters:
[tex]\[
1 \, \text{meter} = 100 \, \text{centimeters}
\][/tex]
[tex]\[
L = 0.38990825688073394 \times 100 = 38.99082568807339 \, \text{centimeters}
\][/tex]
7. Round the length to the nearest centimeter (if necessary):
The length calculated is approximately 39 centimeters.
### Conclusion:
The length of the shortest closed pipe that resonates at a frequency of 218 Hz, given the speed of sound is 340 m/s, is 39 centimeters.
Hence, the correct option is:
C. 39 cm
