Re: How fast does ice get thicker on a lake.

The proper way to deal with this question involves differential equations
with Newton's law of heating/cooling.  However, it is impossible to answer
this question based on the information provided.  To properly answer your
question, we would have to develop a sophisticated model that incorporates
the size and shape of the lake, inventory all of the sources and sinks of
heat, measure the vertical temperature profile, and estimate convective
processes in the lake, just to name a few. We would also have to know a lot
more about the composition of the lake water including its turbidity and
salinity (colligative properties affect freezing temperature), effective
heat conductivity, etc. Measurements like the 10" of ice is not very useful
unless we know how deep the lake is at the place where the 10" of ice was
measured.  You'll find that the thickness of the ice will vary, increasing
near the shallower and stagnant areas of the lake. Also, what is the long
term temperature history of the area?  Large bodies of water can store lots
of heat.  While freezing the top layer may just take a cold snap,
significantly altering its average and deep water temperature takes a
longer period of cooling.

To put your question in a more mathematical form, I think you're asking
what is dz/dt of ice, where z(0) is the surface and decreases with depth. 
The first order equation, neglecting all of the complexity I previously
mentioned, relates dz/dt with the rate of cooling/heating (dH/dt). You
would have to make plenty of assumptions and in the end any scientist would
probably end up pretty embarrassed of how inadequate their model is stacked
up against reality.

A more accurate engineering approach involves using statistics to correlate
the thickness of the ice with the temperature of the ground and air, but
even here, your answer would vary depending on depth of the lake at which
the measurement was made and how deep into winter the cold snap came.

Sorry, but there is no simple answer to this question.