Thursday, July 28, 2011

The Laziness Index

The Laziness Index

By Bobby Neal Winters
I got a call the other day from my old friend Bubba back home.  He was being strangely thoughtful.
I often forget that Bubba received a college education because, I suppose, he so rarely gives any evidence of it.  The fact the of matter is that Bubba could always be quite successful at anything he wanted to do. He never suffered from a lack of IQ; it was more a deficit of “want-to.”
“So,” he began slowly, “you’ve heard of the heat index, haven’t you?”
“Yes,” I replied.  I’d heard a lot about it recently as it had been over 100 degrees and the heat index hand been exceeding that level of five or ten degrees. “We’ve been experiencing it directly.
“Well,” again he was approaching the subject with uncharacteristic care and thoughtfulness, “is it real or just something made up?”
“What do you mean?” I asked.  I am often not very sure where he is coming from, and, though I had an idea, there are times when it is just better to play dumb and ask questions.
“I guess I am asking if it corresponds to reality as described by modern physics or if is just something invented by the pretty talking heads they hire to report on the weather so they can get people to tune in and learn how miserable they are?”  Bubba can be pretty blunt.
“First of all,” I said, “not all of the folks who talk about the weather are all that pretty, and second some are pretty smart.  But to answer your question it is an attempt to capture something real.”
I went on to talk about the affect of humidity.  Humidity can have a tremendous effect on comfort. The human animal uses perspiration as a natural means of air-conditioning.  The sweat glands release water, i.e. sweat. 
When this water evaporates, it absorbs heat and that heat is moved away from the body.  In areas of high relative humidity, the rate of evaporation is slowed because there is a limit to how much moisture air can hold at any given temperature.
“There’s a formula for it, isn’t there?” he asked. “Do you know what it is?”
“There is a formula for it,” I said. “But I don’t know if off of the top of my head.”
“I thought you were a math teacher,” he said, sounding superior now. “What do you folks do if it’s not remembering formulas?”
He does this just to hector me.  In the past, I have attempted to justify my existence both as a mathematician and a teacher of mathematics, but I’ve never gotten anything to stick.  I’ve finally decided that he does this simply to annoy me, so, to avoid rewarding this behavior, I just ignore him.
“We do other things,” I said. “Besides it’s something I can look up anytime on the Internet.”
“What about wind chill?” he asked.  “Is there a formula for that too?”
“Yes,” I said, and, seeing where this was going, I added, “and I don’t know that one either because I can look it up any time.”
In my mind, I was being subtle.  I was attempting to plant the suggestion that, as he spends hours upon hours on the Internet surfing from Sasquatch site to Sasquatch site and finding the perfect fishing lure, he might be able to look this up himself.  I so easily forget that such methods are futile when dealing with Bubba.
“I’m glad to hear that you can look them up any time,” he said, “because I want you to look them up for me and explain them to me.”
I could see that I’d been out-maneuvered, so when we finished our conversation, I went to the Internet and looked the formulas up.
It turns out that, while the formulas are not sophisticated from the point of view of a mathematician, they are complicated from the point of view of the man-on-the-street--or the Bubba-on-the-dirt-road if you prefer. The formula for the heat index can be stated as follows:

The variable T stands for temperature measured in degrees Fahrenheit and R stands for relative humidity as a percent, i.e. 50 rather than 0.50.  The subscripted “a” variables stand in for particular values that I will now give below:

This formula involves no exotic functions, such as logarithms or the various trigonometric functions; it only requires addition, subtraction, and multiplication.  But it contains nine terms and the coefficients--the “a” variables--are decimals and some require scientific notation to represent.  In doing actual calculations with this formula, one would be well-advised to arrange his work carefully.  I myself prefer to put this sort of calculation in a spreadsheet.
The formula for wind chill is shorter, but it is a tad more exotic:

Again the T is temperature in degrees Fahrenheit while the V is wind speed in miles per hour.  The coefficients--the “b” variables--are decimals as given below:

What makes it a little more sophisticated that the formula for heat index is that the V variable sports an exponent of 0.16.  This requires a special key on the calculator to evaluate, so it is no surprise that a spreadsheet is the best place to implement this calculation.
After looking these up and playing with them in a spreadsheet, I did a little report on it for Bubba and sent it off in an email. 
I was also careful to add that the heat index formula was only good for temperatures higher than 80 and the wind chill formula was only good for temperatures lower than 50. I think that may have been a mistake on my part, because it seemed to have turned Bubba’s thoughts in a particular direction.
My cell phone rang and the called ID read “Bubba.”  I took a deep breath, as has become my practice, and answered.
“Those are some durned old complicated formulas there,” he said. “Can’t you come up with anything simpler?”
“They are what they are,” I replied.
“You know,” he was now getting thoughtful again, “The fact that the heat index only works above 80 degrees and the wind chill index only works below 50 degree got me to thinking that these formulas are just ways of telling you when it’s too bad to work outside.”
“That’s not a bad way to think about it,” I said.  Being a teacher, I always try to encourage thinking that is in the right direction.
“That made me wonder whether we might turn that around a little bit,” he said.
I was beginning to get nervous.
“Oh?” was all I could manage.
“Yeah,” he replied.  “I got to wondering whether maybe we out to figure out when it’s to good to work outside.”
“To good to work outside?” I was honestly confused.  “What do you mean?”
“Well, now, if it’s 70 degrees outside with a light breeze and there are a few clouds here and that and the fish are jumping, then I don’t want to waste a day like that on work.  I was wondering if you might could come up with a formula to say that.”
“Yeah, you could factor in the temperature, the percentage cloud cover, whether the fish were biting, the price of beer...”
I cut him off.
“You know what?  I think the weather might be too good even now for me to work on that formula.”
“Maybe I could’ve figured that,” he said.
And we hung up.

1 comment:

Quintopia said...

Any idea as to where those sixth roots in the wind chill formula come from? It's not at all clear to me why they should be...