rate-time-distance problems

Most of the rate-time-distance problems that students find in their Algebra textbooks are contrived and pointless. There's no reason why students should care about how long it would take for two trains separated by 200 miles, one traveling at 45 miles per hour and the other traveling at 65 miles per hour, to pass one another on parallel tracks.

However, current storms and tornado warnings suggest very relevant and much more interesting problems to consider. Meteorologists, or their underpaid underlings, are currently making many similar calculations every minute:

A storm traveling east at 50 miles per hour is currently 32 miles away. How many minutes will it take for the storm to reach your town?

Assuming the storm picks up speed at a rate of 3mph every five minutes, how long will it take to reach your town?

Two storms, one 54 miles to the north and traveling at 52 miles per hour, the other 38 miles to the west and traveling at 45 miles per hour, are headed in your direction. Which will arrive first?

And the list goes on...

a geometric observation

I've been spending a lot of time thinking about centers of triangles recently -- they're fascinating. Here's a recent observation about circumcenters (the point of concurrency of the perpendicular bisectors of a triangle):

The measure of the angle formed by constructing segments from two vertices of a triangle to its circumcenter is twice the measure of the angle at the third vertex.

In other words, given ∆ABC with circumcenter H, m∠AHB = 2(m∠ACB).

A proof? Think about circumscribing a circle about the triangle; the circumcenter of the triangle is the center of that circle...

chocolate oranges

Every year around this time, I open up a chocolate orange and begin to wonder the same thing. As I remove wedges of the chocolate treat, is there any point at which the surface area of the partially-eaten chocolate orange is greater than the surface area of the intact orange? In other words, given a piece of foil that just covers the full orange (with no foil left over), will removing wedges render the foil insufficient to cover the remaining orange?

some sums

Had trouble falling asleep last night, so I began adding up numbers in my head. "Discovered" this interesting property of integers, though I'm sure it already exists more formally in the mathematics world:

Let n be a positive integer. Let S be the sum of all integers from 1 to n, inclusive.

If n is odd, then S is a multiple of n. However, the same property does not exist for even numbers.

It will take me some time to try to even wrap my head around this, and who knows if I even have the ability to do so. But pretty cool nonetheless. And a bit frightening that this is what goes through my head in the wee hours of the morning...

correlation?

I have noticed more white hairs on my head since getting married...

counting birthdays

happy (26th) birthday, nick!

Because math, in so many ways, in grounded in counting, I spend a lot of time thinking about how we quantify things in our lives. I've also been thinking a lot about birthdays recently, as I find myself surrounded by many friends and family members who were born in this tenth month.

What has always bothered me most is that a person can truly have only one birthday, to be celebrated on the day of one's birth. Everything that comes after represents an anniversary of one's birth day. But this is a minor point, an argument that I concede I will never win.

The argument I will win, however, is the following: We almost always incorrectly count birthdays. For example, the date on which one turns 25 is actually one's 26th birthday! But look at the Hallmark cards that arrive in the mailbox and at the writing on the cake and they will all say "Happy 25th Birthday!" Has anyone ever looked at these and said, "Hm... These are a year late."

anti-discounting

A few friends and I came across this parking meter in Amherst, MA a few weeks ago. We have not doctored this image at all.



1. What's wrong with the picture?
2a. If the meter accepted pennies, how many minutes would a penny buy you?
2b. If the meter accepted half dollars, how many minutes would 50 cents buy you?