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the renowned italian physicist enrico fermi@xcite was famous for his ability to make reliable estimates , even with data that would have seemed insufficient to many people .
one of the best known examples of this is the estimate he made of the power of the first atomic bomb detonated on july 16 , 1945 at the new mexico desert , measuring the distance travelled by a few scraps of paper that he dropped to the ground while participating as an observer of the explosion@xcite .
later , when he served as professor at the university of chicago , fermi used to pose such problems to his students as a teaching method .
the most famous was to estimate the number of piano tuners in chicago at that time . making a series of reasonable assumptions , such as the number of people living in chicago , the number of people living on average in every household , every how many houses there is a piano , how often a piano should be tuned , how long it takes for a tuner to do his work , etc .
, he could effectively estimate the number of piano tuners in chicago , obtaining a result that compared reasonably well with those contained in the telephone directory .    in the first year of the physics teacher program at the unrn
we frequently face the students with everyday problems , which can sometimes be solved   la fermi " .
we find it is a very good method to help students sharpen their imagination as well as their observation skills , and at the same time is a training in logical reasoning .
within this type of problem we recently raised the possibility of estimating the power developed by the puyehue - cordn caulle volcanic system , near the argentine - chilean border , in its eruption of june 4 , 2011 at 17:00 utc ( 14:00 local time , gmt-3 ) . to do this
we consider only the first eruption , which covered with ashes and sand the some small villages in chile and the cities of san carlos de bariloche ( @xmath0w , @xmath1s ) , villa la angostura ( @xmath2w , @xmath3s ) and surrounding areas in argentina . in figure[fiaqua ] the first stage of the eruption
is shown , as it has been recorded from space by nasa aqua satellite@xcite .    ]
the working hypotheses that we use are :    1 .   area covered by sand and dust . + according to photographs taken by nasa terra satellite@xcite on june 5 , 2011 , wind conditions during the eruption day made the cloud of sand and ash follow almost a straight line running along the nahuel huapi lake , in the wnw - sse direction .
although the fallout was only partially visible on water , it is clear that it covered the entire area @xmath4 of the lake .
therefore we use this area as a parameter to estimate the amount of ash fall . as a working hypothesis
we assume the area covered by ash and sand to be about three times the area of nahuel huapi lake , @xmath5 .
+    + in figure[terra ] the nasa terra satellite image for 05 jun 2011 at 13:45 utc is shown .
the blue profile corresponds to the coastline of the lake , and in red the area covered by sand and ashes is highlighted . as a guide to the eye , a rectangle of area @xmath6 is also indicated . to determine @xmath7 ,
the following data was used : the surface of the lake is @xmath8km@xmath9 ; adding the surfaces of the largest islands , victoria and huemul , totalling @xmath10km@xmath9 , we find @xmath11km@xmath9 . so the area covered by ash would be @xmath12km@xmath13km@xmath14m@xmath9 .
the picture has been post processed by using the google earth software@xcite .
thickness of the sand layer .
+ in line with the above hypothesis on the covered area , we assign a value to the average thickness of the layer of sand and ash
. it is known that a layer over @xmath15 m covered villa la angostura while east of bariloche the layer was less than @xmath16 m thick . on the other hand reports indicate an accumulation of more than @xmath16 m in the town of ingeniero jacobacci ( @xmath17w , @xmath18s ) 210 km eastward from bariloche .
this corresponds to ash and sand that did not fell into the area @xmath7 but which must be included in the computation since its elevation affects the energy balance of the volcano . to account for the total energy cost we assume an average height of @xmath19 m . covering the chosen area @xmath7 .
duration of the first eruption .
+ the ash fall in bariloche started at 16:30 local time ( art , gmt-3 , 19:30 utc ) and ended five hours later , at 21:30 art , so the total time of the eruption first phase is 5 hours , @xmath20s .
4 .   height reached by the cloud .
+ the height of interest is the difference @xmath21 where @xmath22 m is the height of the volcano .
we have @xmath23 m , an average value as different components of the column reached varying heights , the lightest getting up to @xmath24 m .
density of the mixture of sand and dust .
+ the density of the sand and ash fell in bariloche was determined using household items : a measuring cup used to measure the flour or sugar and a kitchen scale .
the result was @xmath25kgm@xmath26 .
this value , lower than the density of water , is explained because ash and pumice , which floats on water , are mixed with denser components that tend to sink into water .
a simple volumetric calculation shows that the volume of sand and ash fall is @xmath27 considering that a cargo truck can carry about @xmath28m@xmath29 , the amount of sand fall is equivalent to @xmath30 i.e. , more than twenty four million trucks of sand .    an alternative way to visualise the amount of sand consists of calculating the height it would reach if it were placed in a square area of side @xmath31 by stacking a pyramid of height @xmath32 . the volume of the pyramid @xmath33 must match the estimated volume of sand
however , as the angle @xmath34 of inclination of the pyramid should not exceed the `` angle of repose '' of the sand , approximately@xcite @xmath35 , there is a link between the height @xmath32 and the base @xmath31 .
we have the relations @xmath36{\frac{6v}{\tan \theta } } \\   h&=&\frac{l}{2}\tan \theta\end{aligned}\ ] ] with @xmath37 .
it follows @xmath38 this means that the fallen sand could be stacked in a square field @xmath39 km wide ( with an area of @xmath39km@xmath40 ) , making a @xmath41 m high pyramid . as a visual reference we note that the area of the emergent of huemul island , in bariloche is about @xmath42 ha ( @xmath43km@xmath40 ) , and its height certainly less than the @xmath41 m we obtained here .      to determine the mass of sand fall , we multiply the calculated volume by the measured density , to obtain @xmath44 i.e. about 100 million of metric tonnes .      the energy required to raise this mass of sand to @xmath45 m
can be estimated from the potential energy acquired when it reaches its maximum height .
we have @xmath46 where @xmath47 is the acceleration of gravity .
thus we have @xmath48 that is more than one thousand kilotons of tnt equivalent . for comparison , we mention that the 2011 earthquake in japan had a magnitude of @xmath49 in the richter scale@xcite and released an energy@xcite of @xmath50j ( @xmath51kt of tnt equivalent ) .
another reference point is given by the total energy produced in a country .
for example , the installed electric power in argentina@xcite is @xmath52mw and so , the energy produced during a day is @xmath53j .
thus the energy released by the first phase of the eruption is equivalent to that produced in the whole country in @xmath54days .
the magma rises through the conduit of the volcano mixed with fumes of molten rock and water vapour , which results in a turbulent two phase flow .
the velocity of the magma exiting the crater can be coarsely estimated from the energy conservation law , since the flow , at least for the denser components , is ballistic .
the ashes reaching the top of the column probably ascend the last section by convection . for that reason we took an average altitude of @xmath45 m , a moderate value compared to the @xmath55 m or even @xmath24 m that have been reported . neglecting air friction
, the kinetic energy of fluid at the crater must be equal to the potential energy reached in the final ascent , @xmath56 from this it follows that @xmath57 that is , nearly the speed of sound in air .
this high speed of the outgoing material determines its rapid cooling , which explains the amorphous structure and the presence of micro crystals in the ejecta since production of larger crystals requires slow crystallisation .      assuming that the first eruption lasted @xmath58hours , i.e. @xmath20s , it is possible to estimate the power generated by the volcano , which turns out to be @xmath59 by comparing this result with the installed electric power in argentina mentioned above
, the ratio obtained is @xmath60 the power developed by the volcano is @xmath61 times the installed power in argentina .
we can also compare our result with the worldwide installed electrical power@xcite , which is @xmath62tw@xmath63w .
a percentage comparison results in @xmath64 so the power developed by the puyehue volcano is equivalent to about 2% of the global electric power .
in the problem of piano tuners , fermi could compare the estimated result with the answer obtained from a query in the telephone directory .
the discrepancy could be attributed to the fact that some tuners had no phone or were not listed as such in the directory , or to an error in the various estimates .
but this last point is part of the game : the idea is not to obtain an accurate result , but to reach a reasonable approximation . as such
we understand a discrepancy not exceeding a certain limit , say half an order of magnitude , or a factor of five .    in what follows we comment on each of the scenarios we used :    1 .
area covered .
+ in the eruption of june 4 , 2011 ash and sand covered an area larger than the one assumed in this work .
that could mean that we underestimate the energy and power of the volcano .
we tried to compensate for material falling beyond the chosen zone , by assuming a uniform thickness of @xmath65 cm in the reference area .
average thickness of the layer of sand and ash .
+ it is known that in the vicinity of the volcano on the chilean side and in villa la angostura the layer of sand and ash reached @xmath66 cm , while further east it did not exceeded @xmath58 cm .
we believe that the figure of @xmath65 cm distributed uniformly over an area equal to three times the area of the nahuel huapi lake ( over @xmath67km@xmath9 ) is appropriate as an average estimate .
3 .   estimated time of activity . +
the fall of sand and ashes in bariloche ceased at 00:30 utc ( 21:30 art ) , that is the first eruption lasted five hours .
the thunder caused by electric shock lasted a total of approximately @xmath61hours indicating that volcanic activity continued until then .
this does not affect the estimation of the energy developed in the first five hours .
4 .   height reached by the cloud .
+ reports on june 4 indicated a height of the column of up @xmath24 m above sea level .
as the column included different density components , we believe our assumption @xmath23 m for the thicker components of the plume is a reasonable estimate for our purposes .
density of the mixture of sand and ashes .
+ the density measurement was made with a sample collected within hours of the onset of the rash .
samples taken a few days later gave higher density values , probably because rain water dissolved or dragged away part of the components and swept them away .
6 .   output speed .
+ we are not aware of measurements of the output speed of the magma in the volcano being made , but specialised authors have studied the supersonic flow of magma in volcanoes in some typical situations of the so - called explosive pyroclastic eruptions ( see for example @xcite ) , as the puyehue - cordn caulle eruption of june 4 , 2011 .
an eruption of this type occur in the year 79 of the christian era at the vesuvian volcano , inducing the tragic disappearance of the city of pompeii .
this eruption had two distinct phases : first a _
phase , where material was ejected in a tall column , spread in the atmosphere and fell to earth like rain ; followed by a _ pelan _ phase where material flowed down the sides of the volcano as fast - moving avalanches of gas and dust , called pyroclastic flow ( pyroclasts are rock fragments formed by a volcanic explosion or ejected from a volcanic vent ) .
the solution of the present problem indicates that using knowledge normally available to high school students , and introducing a series of reasonable assumptions , it is possible to solve approximately a problem resulting from a natural event that affects everyday life .
andrew w. woods and sally m. bower , _ the decompression of volcanic jets in a crater during explosive volcanic eruptions _ ,
earth and planetary science letters , volume 131 , issues 3 - 4 , april 1995 , pages 189205 .
mitchell , volcanic eruption mechanisms , _ coupled conduit flow and shape in explosive volcanic eruptions , _ journal of volcanology and geothermal research , volume 143 , issues 13 , may 2005 , pages 187203    adriana bermdez y daniel delpino , _ la actividad del complejo volcnico puyehue - cordn caulle
y su impacto sobre el territorio de la repblica argentina _ , http://www.conicet.gov.ar/new_noticias/noticias.php?id_noticia=7192&tipo=6&nota_completa=yes[primer informe ] , conicet , june 15 , 2011 .
adriana bermdez y daniel delpino , _ la actividad del complejo volcnico puyehue - cordn caulle y su impacto sobre el territorio de la repblica argentina _ , http://www.conicet.gov.ar/new_noticias/noticias.php?id_noticia=7295&nota_completa=yes&tipo=6[segundo informe ] , conicet , june 28 , 2011 .
on june 4 2011 the puyehue - cordn caulle volcanic system produced a pyroclastic subplinian eruption reaching level 3 in the volcanic explosivity index . 
 the first stage of the eruption released sand and ashes that affected small towns and cities in the surrounding areas , including san carlos de bariloche , in argentina , one of the largest cities in the north patagonian andean region . by treating the eruption as a fermi problem , we estimated the volume and mass of sand ejected as well as the energy and power released during the eruptive phase . 
 we then put the results in context by comparing the obtained values with everyday quantities , like the load of a cargo truck or the electric power produced in argentina . 
 these calculations have been done as a pedagogic exercise , and after evaluation of the hypothesis was done in the classroom , the calculations have been performed by the students . 
 these are students of the first physics course at the physics and chemistry teacher programs of the universidad nacional de ro negro .