A person hums into the top of a well and finds that standing waves are established at frequencies of 150, 210, and 270 Hz. The frequency of 150 Hz is not necessarily the fundamental frequency. The speed of sound is 343 m/s. How deep is the well?

To find the depth of the well, we need to analyze the standing wave frequencies and use the formula for the speed of sound.

The formula for the frequency of a standing wave in a column of air (like the well) is given by:

f = (nv) / (2L),

where:
- f is the frequency of the standing wave,
- n is the harmonic number (1 for the fundamental frequency, 2 for the second harmonic, 3 for the third harmonic, and so on),
- v is the speed of sound, and
- L is the length of the column (in this case, the depth of the well).

Now, we have three frequencies: 150 Hz, 210 Hz, and 270 Hz. We can see that these correspond to the first, second, and third harmonics respectively.

For the first harmonic:
150 Hz = (1v) / (2L) -----(1)

For the second harmonic:
210 Hz = (2v) / (2L) -----(2)

For the third harmonic:
270 Hz = (3v) / (2L) -----(3)

We can solve this system of equations to find the depth of the well (L).

First, we divide equation (1) by equation (2) to eliminate v:
(150 Hz / 210 Hz) = [(1v) / (2L)] / [(2v) / (2L)]
0.7143 = 1/2
This simplifies to: 0.7143 = 0.5

Next, we divide equation (2) by equation (3) to eliminate v:
(210 Hz / 270 Hz) = [(2v) / (2L)] / [(3v) / (2L)]
0.7778 = 2/3
This simplifies to: 0.7778 = 0.6667

From these two simplifications, we can see that the frequencies are not in the ratio of small whole numbers. Therefore, these values are not the exact harmonics of the same system. This implies that the well is not resonating at its fundamental frequency.

However, we can still use equation (2) to find the depth of the well. Rearranging equation (2), we get:
L = (2v) / (2 * 210 Hz)
L = v / 210 Hz

Substituting the value of the speed of sound (v = 343 m/s) into the equation:
L = 343 m/s / 210 Hz
L ≈ 1.633 m

Therefore, the depth of the well is approximately 1.633 meters.