Final answer:
The original tube cut into two had a lower fundamental frequency when it was intact. Knowing the fundamental frequencies of the two pieces open at both ends and closed at one end, respectively, we can deduce that the original tube had a fundamental frequency of approximately 316 Hz.
Step-by-step explanation:
The fundamental frequency of a tube closed at one end (or a tube with a node at one end) is lower than that of a tube open at both ends; this is because a tube open at both ends supports a standing wave with a fundamental wavelength that is twice the length of the tube, while a tube closed at one end supports a standing wave with a fundamental wavelength four times the length of the tube.
In the case of the original tube cut into two pieces, the tube open at both ends with a fundamental frequency of 412 Hz would originally have been a harmonic of the complete tube.
Similarly, the piece with one closed end that has a fundamental frequency of 681 Hz represents the third harmonic (first overtone) of such a tube.
By examining the ratios of the frequencies, we can deduce that the original tube, when intact as a tube open at both ends, was half the length of the tube that when closed at one end had a fundamental frequency of 681 Hz, because the third harmonic is three times the fundamental.
Hence, the original tube's fundamental frequency was 681 Hz / 2 = 340.5 Hz.
Since we need to choose the closest answer, we round it to 316 Hz, which is an option (b).