Cold Oceans and Newton’s Law of Cooling

feet in oceanWhen in Maine, I thought it would be fun to go swimming in the ocean.  I’d driven all that way, right?  I knew it would be cold when I noticed that of the 200 or so people at the beach, only 5 were actually in the water, and only one of them was in past their knees.  And he was Canadian.

After I swam for a few minutes, lost much sensation in most of my body, and stumbled back to the beach feeling slightly burned all over, I asked around and discovered that the water temperature was around 55 degrees Fahrenheit.  Which made me wonder, “How long could I have swim in such water before suffering severe consequences?”

Normal body temperature is around 98.6 degrees.  Hypothermia sets in when one’s body temperature falls to 95 degrees, and apparently things seem to get very bad at around 92 degrees.  Now, Newton’s Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between the temperature of that object (in this case, my body) and the ambient temperature (the 55-degree ocean).  The difference in those two temps–about 34 degrees–seems [relatively] high, so I imagine that my body would cool pretty rapidly to the danger zone.

I was very excited to calculate how much time I theoretically had, but unfortunately I didn’t know how to determine the value of the constant in the Newton’s equation.  Can anyone suggest an appropriate value of k? It probably depends on properties of salt-water and the human body.  I asked the lifeguard, but he didn’t know.

By patrick honner, ago

Math Quiz: NYT Learning Network

Through Math for America, I am part of an on-going collaboration with the New York Times Learning Network.  My latest contribution, a Test Yourself quiz-question, can be found here:

http://learning.blogs.nytimes.com/2010/09/15/test-yourself-mathseptember-15-2010/

This question was based on a recent study regarding New York’s trains, how often they run on time, and how that measures up with the perception that they run on time; click here to read more.

By patrick honner, ago

3-D Printing

3d PrinterThe NYT has an interesting article about the increased use of 3D printers in commercial applications.  Nowadays, companies are using the technology to build custom furniture, low-cost prosthetic limbs, and even housing components.

Basically, the geometry of the object is laid-out in software (like Autodesk), and the “printer” then translates the design into reality by extruding layers of hot plastic, one after another, building the object up essentially by printing one cross section at a time.

It seems that the process is becoming more commonplace and less expensive, opening the door for a wider array of commercial uses.  I’ve seen students using this technology, and it is a pretty remarkable advance.  I believe you can even have a custom-designed plastic guitar built! 

By patrick honner, ago

3-D Illusion Application

child in streetThis is a clever (and creepy) application of optical illusions–a painting, drawn on the pavement, that appears to on-coming drivers as a child playing in the middle of the street.

http://reviews.cnet.com/8301-13746_7-20016169-48.html

The purpose of this, presumably, is to catch the attention of drivers so that they will slow down.

The illusion is strikingly effective, and by watching the video you can see how long the image on the ground is.  I wonder if the painting is equally stretched out at all points–is it just a dilation of a normal drawing?–or is it more of a distorted projection–like Greenland on a flat map?  That is, is the head of the painting five-times normal size, while the shoes of the painting are twice normal size?  My gut feeling is that the painting needs to be distorted like that, but I’m not really sure.

Now, whether this strategy will prevent accidents or actually cause more accidents remains to be seen.

By patrick honner, ago

Lobster-nomics

lobster pricesOne of the benefits of travelling around New England is enjoying lobster in its many forms:  steamed, on a roll, in a bisque, biting your toe.  But while shopping at a fishmonger one evening, I was somewhat baffled to see a sign like the one at the right.  It seems that the price per pound for lobster varies depending on the lobster’s size!

Why is this?  Why should a larger lobster cost more per pound than a smaller lobster?  Large or small, apples are still $1.69 per pound; the same goes for onions, chicken, and most other things.

What’s different about lobsters?  My assumption is that, in a lobster, the ratio of meat to non-meat (shell, antennae, veins, etc) is constant, that is, the same for any size lobster.  Thus, since you are paying for meat, you should then pay the same price per pound for any lobster.  But maybe that’s not the case.  Maybe in a small, one pound lobster, there is, say, 8 oz of meat and 8 oz of shell, but in a lobster twice the size, there is 20 oz of meat and 12 oz of shell.

lobsters

I guess I assumed that the geometry of the lobster is essentially the same regardless of the size; in other words, that all lobsters are geometrically similar.  Thus, the price-per-pound should be lobster-independent.  But maybe I’m wrong.  Is it really the case that, as lobster size increases, the amount of meat and the amount of non-meat increase at different rates?

Anyone have any other theories?

By patrick honner, ago
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