Tuesday, May 1, 2012

Impossible Khan Academy Question

KhanAcademy.org is very cool.  This Web site offers lessons and exercises on a wide variety of mathematics, science, and other subjects.  I am going through and playing with some exercises in hopes of honing my advanced statistics, trig, and calculus skills.

I came across this scuba diving example that uses simple arithmetic to solve.  The math is correct, but the situation is virtually impossible:
Daniel was scuba diving 28 meters below sea level when he spotted a beautiful fish below. To see the fish up close, Daniel dove 29 meters until he was level with the fish, staring into its eyes.
Where was the fish relative to sea level?
Answer: -57 meters

57 METERS DEEP!? Really? That's 187 feet!  Let's convert this situation to imperial units and interpret what is more likely ocurring:

Some BS open water diver isn't prepared for a deep dive and heads down to 92 feet on a ledge somewhere amazing for far too long time and gets narced out of his mind since he's breathing air instead of Nitrox.  In a hallucination he sees something and decides to dive another 95 deeper and immediately dies from running out of breathing gas.
How deep was Derpo when he died?
Answer: Derpo died at -187 feet

The follow is my nerdy response when I reported a problem for this question:



Greetings,

Although the math is correct, there is a problem with the unrealistic units of depth that are chosen in this example.

A basic recreational scuba diver - who would be interested in observing and following fish rather than performing industrial or scientific work - is generally limited to diving within 100 feet from the surface (≈30.5 meters).  Beyond this depth a number of physiological factors limit a dive without having highly technical training and gear; including nitrogen narcosis, oxygen toxicity, and an elevated risk of decompression illness.

The question states that a diver finds a fish at 28 meters below the surface (92 feet), and descends another 29 meters (95 feet) below this, giving a maximum dive depth of 187 feet.  This diver will surely suffer narcotic effects from the high partial pressure of nitrogen unless he or she were to switch to an anoxic breathing gas that is a specific mixture of oxygen, nitrogen, and helium (referred to as "heliox" or "trimix") which cannot otherwise sustain life at the surface.

Beyond physiological limitations, it is technically plausible however unlikely that a diver will encounter conditions where visibility beyond 80-90 feet of vertical visibility.  In the case that a diver could view an object that lies an additional 95 feet below in the water column with any definition, it is unlikely that the object will appear to be anything more than a dark shadow, as water quickly absorbs much of the visible light spectrum.

Without significant and expert dive planning in perfect conditions, this description of a scuba dive is virtually impossible.  Simply switching the units of depth from meters to feet would make this a far more plausible, and realistic example.

Best regards,
 - Derpington