Ask Marilyn ® by Marilyn vos Savant is a column in Parade Magazine, published by PARADE, 711 Third Avenue, New York, NY 10017, USA. According to Parade, Marilyn vos Savant is listed in the "Guinness Book of World Records Hall of Fame" for "Highest IQ."
In her Parade Magazine column of December 7, 1997, Marilyn could not explain how to use a barometer to calculate the height of a building by measuring differences in air pressure, or to describe any limitations in the accuracy of such measurements.
All three of Marilyn's answers are "lateral thinking" solutions to the problem which is usually posed as a set of successive answers that a physics student gives on a test that asks this question. Usually it is only implied but not stated that the answer that the physics professor wanted, as being uniquely suited to a barometer, is to measure the air pressure at the base of the building, and also at the top, and use the difference to calculate the height. This would need to be done on a day that the barometric pressure is steady; if the barometric pressure is rising or falling between the time that the the two measurements are taken, the results would be in error.
Marilyn seems to neglect this "linear thinking" answer, which is however mentioned in the following two web pages:
By the way, I seem to remember from ground school at La Guardia airport that the appropriate ratio is 1000 feet for each inch of mercury, so if the precision of the barometer is the nearest 0.01 inch (the precision usually given in weather reports), you can expect 10-foot accuracy, or about one floor.
Charlie Kluepfel provided the obvious theoretical answer to the barometer question. His answer does not hold up so well in actual practice.
Barometers were used on the earliest measurements of the Colorado mountains. With several thousand feet of difference to work with, the error was often several hundred feet; in some cases, thousands of feet. In short, a barometer could not tell the surveyor whether one peak was 100 feet higher than another, or vice versa.
A barometer measures only pressure, nothing else. Air pressure can be affected by humidity, temperature, and other factors. It would not be too surprising for a barometer to indicate that the cool roof of an apartment building is lower than the sweltering ground floor!
Combined with a hygrometer, a thermometer, and a bunch of other goodies from a meteorologist's bag of tricks, the barometer may give you some semblance of accuracy. But this will be affected by the time lag from the movement between the two points of reference -- by the time all the instruments have acclimatized to the second location, atmospheric conditions may have changed enough to throw the readings off.
Double the number of toys that you're using, and take two readings simultaneously, after ensuring that the instruments are all calibrated alike. Then you might get some meaningful numbers. Maybe. We're only talking about a few tens of feet, so don't bet your reputation on it.
But that's a whole lot more complex than "using a barometer" -- the original question didn't say anything about "two barometers and a whole mess of other junk," (which is how the CO mountain surveys were done) although, as stated, it doesn't preclude using all this stuff either.
Marilyn's answers, the classical ones, provide a far more accurate measurement than the theoretical "correct" one. Additionally, as the "Famous Court Case" points out, even though her answers are not what a physics professor may have in mind, they are still quite correct.
BTW, Charlie's ratio of "1000 feet for each inch of mercury" is not correct -- that would imply that the earth's atmosphere suddenly drops to nothingness at 30,000 feet, like water at the surface of the ocean. The atmosphere is gaseous, and therefore compressable, so the heght-to-pressure ratio can not possibly be linear. (Side note: mercury, being liquid, has a linear height-to-pressure ratio.)
My numbers are based on the construction of actual altimeters, which actually read air pressure, in aircraft. I've since dug out my old Private Pilot Exam Guide and looked in the section on the altimeter. In describing the Kollsman window, which is a small inset scale which is manually dialled to calibrate the altimeter to the current sea-level pressure, it mentions "A good thing remember is; when you increase the reading in the Kollsman window, the hands will also show an increase in altitude at the rate of 1000 feet per 1.00 inch, or 100 feet per .10 inch, or 10 feet per .01 inch. For example, prior to departure from an airport, you have set the altimeter to 29.92 and the altimeter reads field elevation. You depart this airport and fly to another location where the reported [sea level] pressure is 29.80. When you change the setting in the Kollsman window from 29.92 to 29.80, the hands will read 120 feet lower."
Of course the fluid air's pressure is not linear. But back in high school, when we studied logarithms (which are also not linear), we interpolated using linear interpolation. Later, in calculus, we learned linearization, where a curve is approximated by a line tangent to the curve, which is good enough for small increments, relative to the rate of change in the slope, equivalent to using only one term in x of a Taylor series. In the Private Pilot Exam Guide, they do provide a table showing a standard atmosphere that shows 29.92 inches at sea level, 28.86 at 1000 feet, 27.82 at 2000 feet, 26.81 at 3000 feet, 25.84 at 4000 feet, ... , 20.58 at 10,000 feet, etc., giving an indication of how well linear interpolation approximates the true value. Apparently, 1.06 inches per 1000 feet would be a better linearization at the altitudes we are talking about.
And, although it's been a while since I piloted a plane, the small planes that I did fly did depend on what amounted to an anaeroid barometer to determine how high they were, but considered accurate enough to be called an altimeter. It also takes less time to go from the base of an apartment building to the roof than to get from one mountain peak to another on foot; anaeroid barometers are less bothered by temperature than are mercury barometers; and the small distances involved would be that much less affected by the density difference. But, yes, I agree there's an accuracy problem.
But there are also problems with Marilyn's answers. The string solution requires the additional equipment of the string and ruler. Dropping the barometer requires an accurate stop watch. Neither lowering the barometer on a string nor dropping the barometer over the side would work If the building has setbacks (ledges) on all sides. And we don't know that the superintendent even knows the height of the building or is susceptible to bribery.