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?

Passing Time and Tolls

coinsI enjoy traveling and I enjoy driving, but I don’t enjoy paying highway tolls.  One way I try to diffuse that annoyance is by hoarding coins when I travel, with the intent of paying every toll on the way back entirely in loose change.

I highly recommend the activity–it’s a fun little counting game, it keeps the coins in circulation, and there is a sense of satisfaction that accompanies using 18 assorted coins in a transaction.

But be warned–if you are travelling along the Hudson River in New York state, on I-87, and you decide to get off at Saugerties, one of the toll-booth operators there does not like pennies.  I mean, really, does not like pennies.  Especially more than 40 of them.  Trust me.

More on Buckyballs

buckyball doodleGoogle has a nice doodle celebrating the 25th anniversary of the buckyball.  (A video of the doodle can be seen here.)

“Buckyball” is the informal name of a particular kind of carbon molecule that, geometrically, resembles the geodesic dome made popular by futurist Buckminster Fuller.  They are more generally known as fullerenes (again, after Fuller), and among other things, have recently been detected in space.

Viewed mathematically/geometrically/graph-theoretically, a fullerene is a solid consisting of only pentagonal and hexagonal faces.  There are many different fullerenes–for example, having 20, 70, or 200 vertices–but what’s amazing is that apparently all of them have exactly 12 pentagonal faces.  Only the number of hexagonal faces changes.

Apparently this fact is a direct consequence of Euler’s formula, namely V – E + F = 2, where V, E, and F are the number of vertices, edges, and faces, respectively, in a given solid.  For example, a cube has 8 vertices, 12 edges, and 6 faces; note that 8 – 12 + 6 = 2, just as Euler requires.

Try verifying Euler’s formula for an octahedron!  Then, when you’re done with that, prove the above remark about fullerenes.

Related Posts

Volume, Surface Area, and Benches

My summer of modest carpentry continued with the staining of this unfinished bench:

stained bench

That one-pint can of Black Cherry stain claims that it will cover 75 square feet of surface.  If so applied, how thick would that layer of stain be?  Let’s go with inches first, and convert to microns later.  Or perhaps a more reasonable question is how does the thickness of the stain compare to the thickness of a sheet of paper?

Hopefully someone will figure that out and tell me.

Map Estimation and Arclength

map -- keyWhenever I spend a lot of time driving, navigating, and map-reading, I find myself making a lot of rough estimates of distances.  The process reminds me of how one estimates and ultimately evaluates, using Calculus, the arclength of a curve.

To find the length of a curve, we approximate the curve with a series of line segments.  It’s easy to find the length of a line segment, and so by sacrificing exactness, you turn a hard problem into an easy one.  This is a fundamental technique in Calculus.

I made my line segments equal in length to 40 miles on the map.  Now just add up the lengths of the line segments to approximate the length of the curve.

maps

Each of the seven line segments is (roughly) equal to 40 miles, so the approximate length of the path from Brooklyn to Burlington is 280 miles (not a terrible estimate).

There are plenty of ways to improve the approximation, and the straightforward, but complicated, calculus approach eventually produces the arclength integral.

On the actual drive my approximations weren’t as good, as I was using an inferior distance estimator.

finger estimation

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