Additional Math Pages & Resources

Thursday, September 30, 2010

Vehicle Dimensions

Back 275 blogs ago, I wrote "language is a system for encoding and decoding information so it can be shared ... a language of counting, measurement, shapes and calculation. A language with precise definitions and specialized terms."

When we talk about certain math-related items, it's helpful to use precise terms. Here's an example.

Sunday my wife and I were looking at a car in front of us. She said "Wow! It's fat". I asked if she meant the rear tires, or the body shape and size. She replied "It's got a wide wheelbase." That was an excellent try - wheelbase is a good automotive dimensional term - but in this case it was not the right word.

Here's a Mercedes factory diagram labeled using terms I learned in an earlier automotive career:

Wheelbase (length) is the distance between the centerlines of the vehicle's front and rear axles. Track (width) is the distance between the centerlines of the left and right tires. Notice the front and rear tracks differ on this car. The wheelbase sometime differs from side to side too, although that's rare.

With enough effort, you can describe vehicles precisely with words, but it's often better to provide a diagram which saves words (and translating).

If you are wondering Why all those measurements?, parking garages and ferries use this to decide where to assign a parking place, how steep a ramp you can negotiate, how much to charge, etc.

Here are two dimension diagrams from Mitchell International, where I worked for many years. These are used by collision repair shops to return your vehicle to the proper condition after a crash.
Without this, it's very hard to return a metal cube to its original size and shape. There are several kinds of dimension products, to suit the equipment a shop uses to straighten vehicles. This data comes right off the car - guys like me spent days carefully measuring all these places! If you need good dimensional data, you can buy it from Mitchell.

Now back to my wife's comment - the best way to express her observation in a mathematical, dimensional, automotive sense would be to say "that yellow car's got a wide track."

Wednesday, September 29, 2010

Zip Zap Zip Strap

What do you call these things?

  • Cable Ties

  • Strap Ties

  • Tie Wraps

  • Zip Straps

  • Zip Strips

  • Zip Ties

Here's how patent applications describe them: 

1. A cable tie is moulded in one piece and comprises an elongate flexible strap provided with ratchet serrations on one side and a head at one end of the strap, the head having an aperture provided with a pivoted pawl having teeth which engage the ratchet serrations of the strap when the free end of the strap is passed through the aperture. 

2. A cable tie comprises a head and a strap of plastic material. The head has a transverse aperture formed therethrough and contains a separately formed barb. The barb has a head engaging portion and a strap engaging portion extending at least partly across the head aperture.

Just for simplicity today, let's call them cable ties.

Ties are not just used to fasten cables together. You can attach signs to a fence, hang pipes from the rafters, or  lock the hubcaps to the wheels of your car. Sometimes the police use them as hand cuffs. I'd say these things are nearly as useful as duct tape!

I used some of the giant ones just this morning to repair a torn panel on the seat of my car. Here are 5 of the 6 straps installed under the seat. 

Cable ties come in many sizes - tiny ones just an inch or two long, and giant ones in excess of four feet. Most of them are made of molded nylon (plastic). If you need something to tie things where it's really hot, you might want stainless steel or military-spec ties. Cable ties can also be customized - with colors, special finishes, stripes, your company slogan, etc.

Here are some write-on flag ID ties in assorted colors, from Cable Ties and More Online Store. Just in case you need to buy some today!

Here comes some math!

 Cable ties have ratings which help you to chose which one to buy. Usually you are given the length and tensile strength in pounds. Strangely, the width and thickness are seldom listed.

Tensile strength means the amount of force that an object can withstand before stretching or breaking. (Go here for more info.) Here are two of the ties I used - one straight and the other deformed due to my pulling with all my strength during the installation:

Here's a selection of choices that I found when shopping on the internet:
  • 4 inch 18 pound
  • 8 inch 40 pound
  • 7 inch chrome tie for engine compartments
  • 7 inch 50 pound flag ID head
  • 7 inch 50 pound Tefzel - radiation, acid and heat resistant MilSpec 
  • 8 inch 50 pound releasable
  • 8 inch 100 pound stainless steel
  • 11 inch 50 pound
  • 14 inch 50 pound mounting head #10 hole
  • 14 inch 120 pound
  • 36 inch 175 pound air duct straps
  • 22 inch 250 pound

The ones I used on my car seat are 48 inches long and rated at 250 lbs each. I used 6 cable ties due to the space I needed to span. Does that mean the seat will hold (6 x 250 = 1500)  fifteen hundred pounds? I hope that poor seat never has to find out! But equally I am sure that I won't break through it as I did the skimpy rubber fabric which came from the factory 20+ years ago.

Tuesday, September 28, 2010

I'm weaker than you are!

"I'm weaker than you are!"

This is not the kind of taunting you normally hear at an elementary school. But it's being shouted constantly around the globe as grown-up kids engage in a playground game called The World Economy Currency Wars.

"You're too weak! Too cheap!" We shout back. "Our citizens buy your products instead of our own! Get stronger or we'll put up barriers to keep you out!"

Of course, how you respond to this teasing depends on the team you play for. The finance minister of Brazil recently called everyone else a cheater. He said countries all over the world are meddling in currency trades, in order to lower their own money against everyone else's money. Why? It makes their exports cheaper and more appealing in world markets. This trick tends to strengthen the local economies of exporting countries with low exchange rates.

The problem is, not all countries can have the lowest rate of exchange at the same time! Why? Because we are comparing ourselves to each other.

(Remember Lake Woebegone, where the women are strong, the men are good-looking, and the children are all above average?)

Japan, Taiwan and South Korea have taken action recently to weaken their currencies against the dollar and the euro. The prices of their products were high compared to China's prices. China hasn't done much to let the renminbi float, despite constant complaints from the US. If it went down faster, Chinese products would cost more in the US, and US goods would be cheaper in China.

Only one small problem - if the renminbi went up, then Japanese, Taiwanese and Korean products would appear to be cheaper. Would we switch to buying their stuff, or would we buy more US products?

While we've been jousting with China, we have also let the dollar fall 25% against Brazil's real, thus imports from Brazil (sugar, oil, etc) are no longer competitive with our own products. Travel in Europe gets more expensive for Americans because our dollar is weak against the euro or the pound, but that's ok with our government because it makes our aircraft, software, etc.  cheap for the Europeans to buy.

This is all too confusing without some facts. Facts that real people can easily interpret with the math skills we learned in school.

How can we measure the dollar versus the other major currencies? By plotting the exchange rates over time. These little charts are generated by Google, using data licensed from Citibank. Just go to the search line and type in dollar vs yen to see for yourself. By the way, you cannot order currency or traveler's checks using these values; they are just for comparison purposes.

In order to win the The World Economy Currency War game, you want the blue line (the value of your country's money) to be lower at the right side of the chart (today) than it was at the left side (5 years ago). Got it? Here we go:

Dollar vs Brazil Real         
(we're down a huge amount, which means Brazilian products are now too expensive)

Dollar vs Euro                  
(volatile, but no clear winner at the moment)

 Dollar vs British Pound   
(we're losing versus the British)

 Dollar vs China Renminbi
(gosh, it looks like we are winning this one, why are we complaining so much?)

 Dollar vs Swiss Franc       
(doing ok against the Swiss)

 Dollar vs Japan Yen         
(we are winning, but since we export very little to Japan, this just makes Japanese products expensive and uncompetitive compared to China & Korea)

Dollar vs S Korean Won   
(we are losing here - notice the surge in 2009? it's why Korean cars have been inexpensive in the US)

Dollar vs Russian Ruble        
(we've kept ourselves close to parity with the Russians)

It would be only fair to present how things look to Mr Mantega, the Brazilian finance manager. Here's the Real compared to other currencies. Notice the lines are all going up! No wonder he's complaining.

Real vs Dollar                     

Real vs Euro                       

Real vs Ruble                     

Real vs Renminbi              

I'm no expert on foreign exchange, but it looks like the Brazilians aren't playing "I'm weaker than you are" as effectively as the rest of us are ... they have become the strongest country on the playground!

Monday, September 27, 2010

It's HOT HOT HOT here today

I've lived in San Diego most of my life, and I don't remember it ever being this hot. Whew!

How hot is it? Darn HOT. Or as TV food expert Alton Brown would say, Gosh-darn HOT.

Let's see if we can take all this QUALITATIVE emotional whining and turn it into a math lesson on QUANTITATIVE measurements. Give me the facts and only the facts, etc.


I went to the thermometer in the warehouse.

This measuring instrument sits on a cabinet about 5 feet off the ground, in the shade, in the middle of 30,000 square feet of warehouse space.

The ceiling's over 20 feet, so you wouldn't call the thermometer location unusual. It reads a cool 99.3° F. Humidity 8%.

For metric lovers, that's around 37 degrees.

I grabbed it off the shelf and took a walk!


I walked around the outside of the building, counter-clockwise at a slow pace. I was staggered by the heat!

I'm not talking about a black asphalt pavement. This is strictly white concrete. In 5 minutes the thermometer came up to 106.9° F with 6% humidity - almost 42° C!

For those of you in tropical climates, I sympathize with your high humidity levels. But we are being scorched and all the life is being sucked out of the plants and soil. The Santa Ana winds are blowing off the desert up over the mountains to us.

This is how (and when) the infamous California wildfires start.


We have a couple thousand square feet of unimproved office space upstairs. That means we have a concrete floor, and insulated roof, and nothing else. I waited a reasonable time and got the heck out of there!

The temperature inside was 102.6° F and the humidity had climbed a little to 9%.

It's not the place I want to be working this afternoon! Luckily we rarely go up there except to stash our Christmas ornaments and old Excel Math invoices from 1998.


Our fixed thermometer says 78° F and the 25% humidity is delightful.

Maybe the systems need to be inspected and the files backed up.

If you dial 866 866 (again) 7026 and ask for the computer room you might be able to find me cool, calm and collected.

If you see any fires heading our way, please give us a warning telephone call, okay?

I took this picture during our last wildfire season
 (just before we evacuated the building)!

Friday, September 24, 2010

Friday Sale: 25% off all your problems!

In Excel Math we teach kids how to deal with fractions and percentages and decimal numbers. Let's see how it's done (there are many ways):

First, what is a fraction?

Now what is a decimal number?

Can we convert decimals to fractions and vice versa? Indeed, we can:

What about percentages? Where do they come in?

Any other related concepts we can squeeze in here? Yes, as a matter of fact, how about probability?

What about manipulating these by multiplication? OK, no problem.

Or division? Likewise

I can hear you muttering Gee whiz, it's Friday. Give us a break already!

OK, see you next week.

Thursday, September 23, 2010

Siren Song

A siren in Greek mythology was a bird-woman or mermaid who sang so appealingly it was hard to resist sailing over to hear more. However, if you did that, you'd crash onto some rocks. Here's Ulysses and his crew checking out one of the sirens.

The term siren song came to mean an attractive proposition that was sure to turn out badly if you fell for it. Before we leave this week's series of blogs on NOISE, fog horns and train whistles, I am going to ignore all siren song warnings and share the Victory Siren page with you.

Back in the 1950's it was common to hear air raid siren being tested - not because we were being attacked, but just so if we did face some kind of invasion, or weather disaster, we would know what the warning system sounded like. Most towns had electric sirens on poles. A few large communities had enormous sirens powered by Chrysler Hemi V8 engines!

This  makes the whole siren concept very appealing to car guys - a big mythical powerplant from the Fifties, mounted on a frame in the yard, easily started and revved up - makes so much noise everyone in town knows you have something unusual and enviable. A small group of folks (no doubt closely related to fog horn collectors) find and restore these sirens. Find out more at the Victory Siren website.

Here's one of the sirens, for your listening pleasure. Keep your hand near the volume control.

Siren Song Warning: You might end up shopping on auction sites for a full-size siren of your own!

Here comes the siren math: the Chrysler hemi engines produce 180 horsepower at 4000 RPM. The entire siren unit is 12 feet long and weighs about 5000 lbs.

The siren generates air vibrations at 460 cycles per second (Hz) and 138 decibels measured 100 feet away. That's equal to 30,000 watts from a stereo system! It's twice as loud as a jet being launched off a carrier's catapult.

At full speed, the 22.75" diameter air compressor pushes 2600 cubic feet of air per minute - it comes out through the 6 horns at 400 miles per hour!

The largest and loudest one ever built is sitting on a mountain-top north of Los Angeles. It's not running now, but stories say that when run at full volume in perfect conditions, it could be heard 60 miles away, in Avalon, on Santa Catalina Island. Here's a map to show you the range.

The distance is 100 km or about 62 miles or 211423 cubits (as used in Greek mythology).  

Don't worry about me getting one - my wife has warned me away from that siren song!

Wednesday, September 22, 2010

Measuring and Mapping Noise

After a few foggy days looking at visibility I dragged the blog into the noisy world of horns and train whistles. Today i've chosen the topic of measuring noise.

(This is turning out to be a very tricky thing, highly demanding of concentration and math skills. It's more troublesome than measuring the heat of chili peppers!)

Noise is sound waves (fluctuating air pressure) bouncing down the tiny channels in our ears and striking our ear drums. It's a very complicated thing to measure. One well-known unit of noise measurement is the decibel (one-tenth of a bel). Noise measurement units - indeed the whole process - are specially-adapted to measure only what people hear. That's not quite the same as measuring ALL the actual air pressure changes around us (noise).  Many sounds that might be heard by a dog, or bat, or microphone are not audible to us, so we intentionally filter them out during our measuring process.

We measure instantaneous noise, but average sound pressure is also important. It's a "weighted average" of the instantaneous measurements over time. This cumulative noise is what wears out our eardrums, causing us eventually to suffer hearing loss.

Sheesh. Even as I write this I realize what a complex subject this is. Probably too much for elementary mathematics, but ... if you want a nice overview of the decibel, you can read about it on Wikipedia.  Here's a preliminary definition:

decibel (dB) is a logarithmic unit that indicates the ratio of a physical quantity (power or intensity) relative to a specified reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities. Because this is a ratio of two measurements of a physical quantity, it is a dimensionless unit

Well, of course it is. Hmmm (I hummed quietly).

Okay, let's assume we did measure some noise - how could we best display our findings?

1. Measure noise in lots of places, over a long period of time, calculate the average noise levels and spread your findings on a map - you would have this plot of the noise around San Diego airport.

The airport is the yellow area in the center. Other colors around the map reflect industrial buildings, parkland, homes, etc. If you live outside these areas then airport noise is an insignificant part of your average daily noise intake (are there minimum daily requirements?).

2. Use special software to take those same measurements and you might get a three-dimensional, computer-generated noise map. This one came from the China Daily newspaper. It shows a neighborhood in Shenzhen China.

3. Here's an interesting way to show an aircraft's acoustical footprint. The odd shapes visually portray the noise of planes landing and taking off.

Patterns in orange represent noisy aircraft - they are no longer aloud (sorry, couldn't resist) in the US. Blue patterns are quiet planes. Numbers above each graphic are square miles of noise impact, and percentage of the aircraft landing at Chicago O'Hare; numbers below are plane model and passenger seating capacity. This presentation shows you to identify planes making the most noise per passenger. Of course you have to do this for each and every variant of aircraft. Not all are shown here.

4. Here's an FAA chart with two different plots. The rear (reddish) shows the number of people flying [passenger emplanements?!]. The front (greyish) shows the number of people affected by excessive airport noise. All the numbers are in millions.

Over 30 years, the number of people exposed to aircraft noise diminished rapidly as passenger numbers have increased gradually.

5. No more. Going any deeper into this noise business will take more mathematics than we can teach in Excel Math...

Tuesday, September 21, 2010

Fog horns and train whistles

Fog horns make Loud Noises, like Bwaaaah-uh!

Here's a big fog horn on Flat Holm Island off the English coast. This photo shows the enormous horn that allows air to escape in the form of low-frequency vibrations (air waves).

What isn't shown are the air tanks and the compressor which generate the sound waves. The two tanks for this horn are each about the size of a car!

Here's the Portland Bill in Dorset, England. My wife and I visited a few years ago. I found this nice recording on YouTube, so you can hear what it sounds like. Technically this is called a diaphone.

Ships also need to have fog horns, as they are moving through the water amidst other ships and all need to know where each other are located.

Finally, this photo shows a train horn. A Nathan Airchime. This baby produces enough noise to awaken sleeping drunks near train tracks (and everyone else sleeping within miles).

Even though I live several miles from the tracks (as the crow flies), we can hear trains at night.

If you are of a horn-blowing, noise-loving nature, check out air horns here.

If you don't like noise and are getting too much of it, check out occupational hazards of noise here.

Tomorrow we'll get to the math of noise and the instruments used to measure it (sound level meters and noise dosimeters).

Monday, September 20, 2010

Visibility and Audibility

The last few blog posts have dealt with reduced visibility due to fog. It's not so foggy today, but I did hear foghorns blowing this morning. They made me think about other ways to communicate when visibility is poor.

Lots of ships sail in and out of San Diego harbor. Like most seaports, we have a lighthouse to send a visual signal to ships. The old historic lighthouse on top of Pt. Loma is a well-known symbol of San Diego. But it was hidden in the fog all the time, so a new one was built closer to the water.

But what if you can't see the light? What do you do? A sailor on a US Navy destroyer told me yesterday that this week they anchored off the coast for a few hours until the fog thinned out enough for them to enter the harbor safely. Notice the fog bank behind the ship in this picture.

Although they have radar, sonar, digital charting, etc., any kind of mishap with a 400-foot destroyer is a career-ending event for the captain and navigators. A short respite at anchor is no trouble at all in comparison!  Now back to my first question:  

Q. What do you do if ships can't see your lighthouse (or land) in the fog? 
A. You blow a fog horn. 

Here's a photo taken while we were crossing the Golden Gate Bridge in San Francisco. Notice the bridge wires quickly becoming invisible in the fog.

 Click here to hear the Golden Gate's foghorn.

Q2. How far can you hear a foghorn? 
A2. About 6 miles (the best answer I could find today). 

We'll do some more fog horn math tomorrow. In the meantime, you can plan a sound vacation here!

Friday, September 17, 2010

After some reflection on visibility ...

It's still foggy this morning, so we might as well continue on the theme of visibility. Yesterday I said we would calculate the optimum eyeball height to see reflective clothing. I think the angle specified by 3M will equal the height of the eyes of a driver,  a few feet above the film and the headlights.

The 3M Scotchlite™literature says the optimum viewing angle is 0.2 degrees at 800 feet.

Let's make a right triangle whose adjacent side is 800 feet and whose angles are .2, 89.8, and 90 degrees. Remember that the sum of the angles of any triangle equal 180 degrees.

OK, there's our triangle. A is the height of the eyeballs, B is 800 feet, and C is the hypotenuse.

A few among us might recall the term Pythagorean theorem but can you remember what it says? Something about A² + B² = C² isn't it? Yes, but the real value is if you know a couple of the sides and angles of a triangle, you can discover the rest of them. We know all 3 angles and 1 side, so we can learn the lengths of the other sides.

This slightly above elementary math, but let's go ahead and calculate the height of A anyway. If you recall the formulas you can do this manually with a pencil, or a calculator (or go here to use a triangle side/angle calculator) then come back and check your answer against mine.

I get the height of A as 2.8 feet which is about 34-35 inches.

I needed someone to measure, so I drafted Darcie who does our order processing. We went out and measured her car's headlight height, and the difference in height from her headlights to her eyes when (1) standing up and (2) seated in the car.

1. Darcie is 5' 7" tall and her eyes are 37-38" above the headlights (depending on shoes!)
2. When seated in the car, her eyes are 48" from the ground; 24" above the headlights

What does all this measuring tell us? Reflective film works best when people are wearing it are within 200-300 yards from an oncoming vehicle. The majority of the light is reflected straight back at the driver, and the long range allows plenty of room to make decisions about turning or stopping.

The safety value of this kind of night-time attire was reinforced last night, when  3 girls walked out across the road in front of me in the middle of a very dark block. They had on regular clothing and were nearly invisible at less than 100 feet. They could see us coming but we almost didn't see them!