In an experiment to determine the velocity of sound in air at room temperature using a resonance tube, the first resonance is observed when the air column has a length of 20.0 cm for a tuning fork of frequency 400 Hz is used. The velocity of the sound at room temperature is 336 m s$$^{-1}$$. The third resonance is observed when the air column has a length of ______ cm.

We need to find the length of the air column at the third resonance in a resonance tube experiment.

For a closed organ pipe (resonance tube), the resonance lengths are given by $$L_n = (2n - 1)\frac{\lambda}{4} - e$$ where e is the end correction. Equivalently, $$L_1 = \frac{\lambda}{4} - e$$, $$L_2 = \frac{3\lambda}{4} - e$$, $$L_3 = \frac{5\lambda}{4} - e$$.

The wavelength is calculated as $$\lambda = \frac{v}{f} = \frac{336}{400} = 0.84 \text{ m} = 84 \text{ cm}$$.

Using the first resonance, $$L_1 = \frac{\lambda}{4} - e$$ and substituting $$20.0 = \frac{84}{4} - e = 21 - e$$ yields $$e = 1.0 \text{ cm}$$.

Finally, $$L_3 = \frac{5\lambda}{4} - e = \frac{5 \times 84}{4} - 1.0 = 105 - 1.0 = 104 \text{ cm}$$.

The answer is 104 cm.