A 'sound' problem

An interesting and practical approach to this problem is to use approximations which can be fine-tuned to reach any desired precision.

We start by estimating an initial value for the depth: $h_1=\frac{1}{2}·g·t^{2}=\frac{1}{2}·9.8·3.2^{2}\approx 50\,\text{m}$ We use this value to estimate the time the sound needs to travel up the well to reach our ears: $t_s=\frac{h_1}{v_s}=\frac{50}{340}\approx0.15\,\text{s}$ We now use this value to correct the falling time: $t_{f}=t-t_s=3.2-0.15\approx3.05\,\text{s}$ This time, the value of $h$ obtained is already good enough for the precision required: $h_2=\frac{1}{2}·g·t_f^{2}=\frac{1}{2}·9.8·3.05^{2}\approx 46\,\text{m}$ We can repeat the sequence to improve the result if necessary.