How are your pipes?

I've talked this week about blood, breathing, snoring, and so on. Today we look at another disgusting topic - arteriosclerosis - material collecting on the inside of pipes, so liquids cannot flow through. Only in this case, it's not MY arteries we are concerned with, but the pipes in my house. I've already had my arteries cleaned.

The house is almost exactly the same age I am. We've had it for 20 years and since only two of us now live it, the house is aging gracefully. But! Earlier this week the sink in our main bathroom stopped draining. We had already tried all the normal (and safe) tricks - boiling water, vinegar, sticking a clothes hanger in the pipe, etc. NOTHING was going down.

Here's a diagram for the non-plumbers, and people in other countries who do plumbing differently.

I took off the trap. I couldn't get anything to go into the drain extension! And I couldn't undo it, although I did discover it was leaking through pinholes. So I squirted the joint (under the trim ring) with liquid un-wrenching juice.

Then I went up on the roof with a plumber's snake, and ran it 30 feet down the vent stack. I got a handful of stuff out. I went back down, and with a torch, giant wrench, pliers and a hammer, pulled pipes out of the wall. The wall pipe was completely clogged! You can see a drain extension pipe here. Notice the hair and the solid gunk that we had to chisel out of the pipe!

Here's where it goes into the wall, after I have spent a 10 minutes scraping it out with long screwdrivers, the plumber's snake, and my fingers. Yuk.

Finally, here's another look at the pipe. It was 90% obscured when I started.

 

So what do you suppose is the problem? Can we define it with math? Let's try. A water conservation site estimates tooth brushing, face washing and shaving all take about a gallon (each):

87,600 = Number of times people brushed teeth in this sink (assume 2 people x 2 times/day x 60 years)

43,800 = Number of times people washed their hands in this sink  (assume 2 times/day x 60 years)

15,600 = Number of times people shaved in the sink (5 times/week x 60 years)

  1,440 = Hand Laundry (once a month x 2 gallons x 60 years)

     200  = Amount of cleaning solutions, soaps, cleansers, etc down the drain (8 oz/week x 60 years)

148,640 = let's round this up = 150,000 gallons of mucky water down the drain.

Is it normal for sinks to clog? I checked a plumbing website and found this diagnosis:

In bathroom sink drains what sticks is a combination of face cream, shaving cream, toothpaste, and stray hairs. That stuff often gets caught on the pop-up and the lift lever just below it, but after years it begins adhering to the walls of the pipes. I've seen it accumulate to the extent that even the trap pipes are almost closed up. 

We had completely closed-up pipes! It's normal. So I took them out and put in some new ones. Aren't they pretty?

I can now expect a few trouble-free years and tens of thousands of gallons to flow down these pipes.

Not So Loud, or so seldom

Yesterday I mentioned a common personal health-related condition - a very unpopular practice that disturbs other people in your house.

SNORING
But sometimes I snore, and so it's an appropriate topic for this blog (if I can discover some math about snoring). How about some numbers about the frequency of this irritating habit?

Perhaps 30-50% of middle-aged or older people snore, compared to about 5-6% of children.

The most reliable methods to control snoring are lose weight, stop smoking, breathe through the nose, and sleep on your side. I have taken the last two of these as my preferred approach. Even though you are asleep most of the night I can tell you from personal experience that it is possible to train yourself to stop snoring by sleeping only on your side, and to keep your mouth shut to breathe through your nose.

Other approaches you might consider include:

• Chin straps to keep your mouth closed
• Nasal strips and nose clips
• Mouth guards
• Fancy pillows
• Surgery
• Drugs
• Ear plugs (not for you but for your family!

SLEEP APNEA
A related condition is called sleep apnea. You stop breathing for a long time (at least 10 seconds),  oxygen levels drop 3-4%, carbon dioxide levels rise, then the "stay alive" reflex kicks in and the body gasps for air to catch up.

People with sleep apnea rarely get enough quality sleep at night, so they are liable to fall asleep in the daytime, even while driving or talking. I have at least 6 friends who have been diagnosed with serious sleep apnea -- it was terrifying to be with them in the car.

The "Medical World" has only known of this condition for about 40 years. About 9-10% of men are thought to have have sleep apnea, compared to about 4% of women. Only 10% of sufferers have been diagnosed. Sleep clinics identify the condition by having you come in and sleep in their observation rooms, or by giving you a recording device to wear while sleeping.

This condition is taken very seriously by doctors, because sleep apnea sufferers have as much as 30% higher risk of heat attacks and a 240% increased risk of congestive heart failure. One study suggested that medical costs for undiagnosed sufferers of sleep apnea might cost twice as much on average (per year) than the cost of a non-sufferer.

The Continuous Positive Air Pressure (CPAP) machine is nearly 100% effective in reducing sleep apnea. The CPAP machine was invented about 30 years ago. ResMed, one of the world's largest suppliers of CPAP machines, is located just a few miles from our offices. I saw an article that stated they probably supply 30% of the machines in the US, and perhaps 50% of those around the rest of the world. It's more than a one billion dollar business for ResMed alone!

Although it works to reduce the effects and dangers of sleep apnea, the CPAP machine isn't a cure, and it's a nuisance. It's certainly not attractive to wear the face mask. At least 25-50% of the people who get the machines stop using them.

Note: If you fall asleep while reading this abstract, you don't necessarily have sleep apnea, you are just normal:

"Recent studies have uncovered high prevalence of undiagnosed sleep-disordered breathing, and its linkage to metabolic or cardiovascular disorders which represent increasing health hazard. However, the mechanistic links behind these disorders as well as their contribution to the experimental observations and treatment responses remain poorly understood. Therefore, the screening of clinical measurements still relies upon relatively simple diagnostic features, such as signal averages or event frequencies, which may represent suboptimal or surrogate markers of the underlying abnormality. Consequently, most patients are being treated with general therapies regardless of the cause of their key dysfunction. Combining experimental measurements with mathematical modelling has the potential to provide mechanistic insights into the individual factors underlying the disease progression, which may finally enable tailored treatment alternatives for each patient. This review depicts a number of modelling approaches to elucidate sleep-related dysfunctions of the human respiratory system, and how these models are being used to translate the measurements first into new ideas and then into testable hypotheses. Such model-based investigations can provide systematic strategies towards better understanding, predicting or even preventing these dysfunctions. Along with the brief description of the modelling approaches, we discuss their relative merits and potential implications especially for clinical research."

Not so fast, or so far, or so often!

"I spoke in error. Apparently there aren't as many miles of blood vessels in the body as I said yesterday. "

Have you ever had to make a statement like this? Or are you in the kind of business that keeps a stiff upper lip and ignores any misstatement, mistakes or misses?

There's no way to know which is really the best policy. In my case, I hate to make mistakes but I'm in a job where we produce a product used by millions of kids, and unlike most books, ours are read from cover to cover and each problem is solved and re-solved many times over. The mistakes pop up like helium balloons escaping from a birthday party.

As far as the blood vessels go, the number is an estimate. More research which I initiated by myself, has shown me that the textbook authors disagree - and why not, they're just people like me. Some say 100,000 miles; others 100,000 km (62,000 miles).  They're all guesses. That's my guess, and I'm sticking with it until I find someone with a blood-vessel-odometer.

And since I am not a doctor or an anatomist, I have nothing to lose by admitting I did not do enough research to be confident of my claims.

However, I have exhaustively researched the following fact: my wife takes about 14 breaths a minute. She's a PE teacher; has been teaching for 30 years, and she knows her body statistics.

My Math expertise tells me that 14 bpm x 60 x 24 hours is roughly 20,160 breaths a day. That's more than 7.4 million a year!
 
My wife is now 56 1/2. If she has taken 14 breaths per minute all her life (on average), how many times has she breathed?

Here's the calculation  56.5 x 365 + (56 ÷ 4 = 14 leap year extra days) x  20,160

Restated, that's ((56.5 x 365) + 14) x 20,160 = 20636 x 20,160 = 416,021,760 or roughly 400 million breaths!

Whew. I know we can control how fast we breathe, and how slowly we breathe (within limits). We just can't control how infrequently we breathe (hold our breath) because eventually our bodies take over if they need more air. Gasp!

Since I'm 3 years older, I've breathed at least 20 million more times than my wife has. I think I'll go lie down now, I'm exhausted. And to keep from snoring, I'll breathe only through my nose!

Who knows how the blood flows?

My fingers are cold. It's poor circulation, I think. Then I wonder - how much circulation should I have?

It's easy enough to learn that the average person should have 5-6 quarts (4-7-5.5 liters) of blood.

I know that my pulse rate is about 78 heart beats per minute.

According to some gory medical websites, every minute the blood in the body travels out to the ends of our fingertips and toes, and back to the heart and lungs. It's estimated that the combined length of all the blood-containing pipes (arteries, veins, etc.) in the body is about 100,000 miles. [who decided that - it's 4 times around the world!]

So the rate of flow is around 5 liters a minute, or (60 x 24 x 5 = 7,200) about 7000 liters a day.

A liter of blood weighs about a kilogram. So your heart is moving 7000 kilograms of blood in a day, up and down in and out your thousands of miles of arteries and veins.

That means 7000 x 2.2 lbs = 15,400 pounds of blood moving around in 24 hours!

No wonder my fingertips are cold! I must be exhausted from all the hard work my heart is doing.

Some people are quite comfortable with the idea of talking about blood. It's not my favorite subject, as I tend to faint at the sight or mention of needles. Blood tests are a grueling experience and I avoid them whenever possible. Why hasn't the medical profession found another way to test people? 

The only time I tried to donate blood, back when I was in college, I was able to eke out about a cupful in 45 minutes - all the other donors had lain down, given blood, eaten a donut, drank some orange juice and were on their way in 15 minutes ... that's me, below:

I lived in the UK for a year during the period of time when "mad cow disease" was discovered, so I have been banned for life as a blood donor to the Red Cross. That saves me from appearing selfish or cowardly if asked to give blood ...

In some parts of the world, cattle can get an infectious, fatal brain disease called Mad Cow Disease. In these locations, humans have gotten a new disease called variant Creutzfeld-Jacob Disease (vCJD) which is also a fatal brain disease. Scientists believe that vCJD is Mad Cow Disease that has somehow transferred to humans. Evidence from a small number of case reports involving patients and laboratory animals suggest that vCJD can be transmitted through transfusion. There is no test for vCJD to protect the blood supply. This means that blood programs must take special precautions to keep vCJD out by avoiding collections from those who have been living where this disease is found.

I think I better go lie down. I'm feeling queasy!

Are you an Excel Math fan?

In the English language a single word can have many different meanings. Today we look at the FAN - a powered device for moving air.

 

FANS and MATH go together, and that's not just a bunch of hot air!

These fans are sitting around inside the Excel Math company offices:

We have floor fans, heater fans, fans with sanitizers, rotating and oscillating fans, etc. Some have vertical shafts around which the blades are set (the ones on either side of the photo) or horizontal shafts (the 2nd and 4th heater-fans). The middle fan with the blue lights is intended to kill germs.

The three middle fans are axial - meaning the blades are attached to a rotating shaft and air flow is in line with the shaft. The tall ones are centrifugal fans. On a centrifugal fan the air does not go in the direction of the rotating shaft.  Their blades are arranged in one of these ways:

I found lots more fans outside the company offices, in our warehouse:

How do we use math to understand fans?

Special units have been created to describe the air moved by fans. Because air can vary in many ways (temperature, density, humidity, particulates, etc.), people in the fan business have defined a set of standard air characteristics:

Standard air is clean, dry air with a density of 0.075 pounds mass per cubic foot (1.2 kg/m³), at barometric pressure of 29.92 inches of mercury (101.325 kPa) at sea level, at 70°F (21°C).

Using standard air allows companies to develop fans that compete on equal terms to move the maximum amount of air with the minimum of noise and power. Sorry, you can't design a fan with only elementary math, you have to be able to understand formulas like Specific Fan Power (how many kilowatts of electricity it takes to move some air):

If you want to know how loud some moving air is, you should understand sones, which describe the noise of moving air. (See my blog on horns and whistles if you like noise!) Here's a sones formula:

Of course, people who don't know math can still move air, and they can do it without electricity ...

If you were thinking about a different kind of FAN - educators, parents and kids who like Excel Math, that's a great use of the word FAN too.

These FANs happily follow and promote the activities of another person, or team, or company. Luckily for us, many educators and kids are fans of Excel Math:

Excel Math fans like to share their thoughts about our curriculum. Go here to read what they say or go to our Facebook page.

More Housecleaning, on a rainy day

Here's where we left off yesterday - trying to clean my computer, repair my crashing iPhoto software and fix up permissions on my damaged drive.

Capacity : 232.6 GB (249,715,376,128 Bytes)

Available : 121.5 GB (130,462,179,328 Bytes)

Used : 111.1 GB (119,253,196,800 Bytes)

Number of Files : 868,750

Number of Folders : 203,911

I'm happy to report the iPhoto crashing has been repaired, and the damaged drive has been fixed too. Now we just need to run a capacity report and see where we have ended up! Do I have more clean space?

Capacity : 232.6 GB (249,715,376,128 Bytes)

Available : 116.3 GB (124,861,341,696 Bytes)

Used : 116.3 GB (124,854,034,432 Bytes)

Number of Files : 916,206

Number of Folders : 205,894

Alas, this looks like more data rather than less! How did that happen? It's like going to a garage sale in your neighborhood - you take things over to sell and come back with more than you took ...

I can tell you what happened. I have an external drive and I moved a lot of things over there, thus cleaning the drive inside the computer. But my iPhoto database (27 GB and 50,000 photos) was on that external drive. In order to rescue the data, I had to move all the photos to my internal drive and create a new database.

I decided I don't want to move the photos back to the external drive (just for the blog clean-up) because it's slower and would take me more than an hour.

Now a question! Did you notice in the statistical report above that I have used almost exactly half of the total space on this drive?

Available : 116.3 GB (124,861,341,696 Bytes)

      Used : 116.3 GB (124,854,034,432 Bytes)

Both say 116.3 GB but when you look at the detailed figures, there's very slightly more available than was used. Do you think I could create a file exactly the right size to split the difference?? Let's see:

 124,861,341,696
-124,854,034,432

we can lose the 1248 at the front to simplify things a little bit

 61,341,696

-54,034,432

   7,307,264

So I need to create some thing(s) that use 7,307,264 bytes. How do I do that? I looked for a file of the right size and copied it. That was too big, so I took it off (emptied the trash) and tried a few others. I got to this point:

Available : 116.3 GB (124,857,974,784 Bytes)

        Used : 116.3 GB (124,857,401,344 Bytes)

 974,784
-401,344
 573,440

Now I need only 573,440 bytes. A little more trial and error, and I came up with this:

Available : 116.3 GB (124,857,679,872 Bytes)

      Used : 116.3 GB (124,857,696,256 Bytes) 

That's even closer to the middle. Nearly perfect, in fact, when you think in computer terms.

 96,256
-79,872
 16,384

I can't seem to get any closer than 16k (16 x 1024 = 16,384) because that's a sector on my hard drive and even the smallest file takes up that amount of room. And if I keep fooling around, caches and buffers fill up, emails sneak in, and my count gets wacky.

So this is how you use your garage sale and elementary math skills to entertain yourself on a rainy day!