## Posts tagged ‘airplane’

### Thinking > Symbols

As if having to wear a mask while flying isn’t scary enough, last week I had a harrowing experience.

It’s normally a three-hour flight from Yellowknife in the Northwest Territories to Portland, but about 20 minutes into a recent flight, there was an unsettling noise followed by an announcement from the pilot: “Ladies and gentlemen, this is your captain. Please don’t be alarmed by the noise you just heard near the left wing. Let me assure you that everything is okay. We’ve lost an engine, but the other three engines are working just fine. They’ll get us to Portland safely. But instead of taking 3 hours, it’s going to take 4 hours.”

She said this matter-of-factly, as if she were commenting on the color of my shirt or pointing to a squirrel that was enjoying an acorn. It was reassuring that she didn’t seem worried — but still, we had *lost an engine*! The cabin grew quiet as we waited to see what would happen next. But, nothing. We continued to fly as if nothing had happened, and the din of conversation slowly resumed.

And then there was another noise near the right wing, followed by the pilot: “Ladies and gentlemen, this is your captain again. Unfortunately, we’ve lost another engine. But the two remaining engines are functioning properly. We’re going to get you to Portland safely, but now it’s gonna take about 6 hours.”

Needless to say, you could hear a pin drop as we passengers waited with baited breath. And sure enough, there was another noise, and the pilot again: “Ladies and gentlemen, we’ve lost a third engine, but we’ll make it safely to Portland with the one engine we’ve got left. It’s just gonna take 12 hours.”

That was when the guy in the seat next to me lost it. “I sure hope we don’t lose that last engine,” he yelled, “or we’re gonna be up here **all day**!”

That story isn’t true. None of it. Not. One. Single. Fact. I mean, come on! Yellowknife? Why would I need to go to Yellowknife? To brush up on my Dogrib?

But the following story *is* true.

My honorary niece Ivy occasionally calls for help with math. Not surprisingly, the frequency of calls has increased with online learning during the pandemic. This is a modified version of a question about which she recently called:

A plane traveling against the wind traveled 2,460 miles in 6 hours. Traveling with the wind on the return trip, it only took 5 hours. What was the speed of the plane? What was the speed of the wind?

It’s assumed that the plane flew at the same average speed in both directions and that the wind was equally strong throughout. On a standardized test, those assumptions would need to be stated explicitly. But on this blog, I make the rules, and I refuse to add more words to explain reasonable assumptions without which the problem would be impossible to solve.

As presented to Ivy in an online homework assignment, the problem stated that *x* should represent the plane’s speed and *y* should represent the speed of the wind. My first question was, “Why? What’s wrong with *p* and *w* as the variables for plane and wind, respectively?” But my second, and perhaps more important, question was, “Why are they forcing algebra on a problem that is much easier without it?”

Somewhere, many years ago, someone decided that the methods of substitution and elimination were critically important, if not for success in life then at least for success in high school. Anecdotally, this seems to be true; I was a bad-ass systems-of-equations solver in high school, and I was also the captain of the track team that won the county title, so… yeah. Sadly, that someone seemed to put less stock in a slightly more critical skill: thinking.

Here’s the deal. Against the wind, the plane covered 2,460 miles in 6 hours, so the plane’s speed on the way out was 410 miles per hour. With the wind, the plane covered those same 2,460 miles in just 5 hours, so the plane’s speed on the way back was 492 miles per hour. Since the wind is constant in both directions, the unaided speed of the plane must have been 451 miles per hour, exactly halfway between the against-the-wind and with-the-wind speeds. Visually,

The top portion of that diagram represents the plane flying against the wind; the bottom portion represents the plane flying with the wind. This makes it obvious that the the unaided speed of the plane is equidistant from 410 miles per hour and 492 miles per hour, and that distance is the speed of the wind.

Algebraically, the problem might be solved like this…

which is no different than the logical approach above: *x* ‑ *y* is the speed of the plane against the wind, and *x* + *y* is the speed of the plane with the wind, so the numbers are obviously the same. But using algebra just seems so much more… *cumbersome*.

Students in elementary classrooms often explain their thinking with drawings and diagrams. Unfortunately, some of that great thought is abandoned when students take a course called Algebra, and the mechanical skill of symbolic manipulation replaces the mathematical skill of logical reasoning. That’s a tragedy. As Jim Rubillo, former executive director at NCTM, used to say, “We have to stop being so symbol‑minded.”

### The Math of Disney

Just returned from a week in the Magic Kingdom, where I learned a lot. Like this tidbit:

A recent government study just confirmed that six of seven dwarves is not Happy.

That’s so good, it deserves a graphic:

I also learned that relativity is a novel concept for kids who are 6 years old. When asked how fast we were traveling on The Barnstormer junior roller coaster, one of my sons replied, “It felt like we were going 100 mph!” The other said it only felt like 50 mph. When asked how fast we were traveling in the airplane on our ride back to Virginia, one son suggested 10 mph, the other suggested 20 mph.

Depressingly, the ratio of bottles of Coke to bottles of water consumed at Disney is almost 6 to 1. According to a Walt Disney World fact sheet, tourists to Disney consume 75 million bottles of Coke and 13 million bottles of water annually. My sincere hope is that most folks use refillable water bottles, which would explain the discrepancy in sales.

And finally a math question.

The fact sheet states that the Earffel Tower, which is the water tower at Disney’s Hollywood Studios, would wear a hat size of 342-3/4, although another reference says that the hat size would be 342-3/8. What is the approximate radius of the Earffel Tower?

### Top Flight Math Jokes

I’ve used the following joke as the opening for many local presentations:

I’m so thankful that I was able to drive here this morning. I’ve been flying a lot for work recently, and last week I had a really horrible flight. I had just finished four long days at a math conference, and I was exhausted when I boarded the plane. We took off, and as soon as we levelled out, I put my head back and closed my eyes.

And then… a thud.

Now, I don’t know if you’ve ever heard a thud while on an airplane, but I didn’t particularly like it. Immediately, the pilot’s voice came over the loudspeaker. “Ladies and gentlemen, this is your captain speaking. I know you heard that sound, and I want to assure you that everything is all right. We’ve lost an engine, but we can still safely make it to our destination with the three remaining engines. However, instead of the flight taking 3 hours, it will now take 4 hours.”

This was unsettling, but after 20 minutes of smooth flying, I closed my eyes once more.

Thud!

Again, the pilot’s voice. “Ladies and gentlemen, it appears we’ve lost a second engine. Let me assure you, we will still make it to our final destination safely, but instead of 4 hours, it will now take 6 hours.”

We then flew smoothly for another 30 minutes… but I was unwilling to close my eyes again.

Thud!

“Ladies and gentlemen, we’ve lost a third engine. We can make it safely with just one engine, but it will now take us 12 hours to reach our final destination.”

Upon hearing this, the guy next to me leaned over and said, “My gosh! I sure hope we don’t lose that fourth engine, or we’ll be up here all day!”

I use that joke because I think it’s funny, but also because it allows me to ask this question: “Consider the pattern. When there were 4 engines in use, the flight was supposed to take 3 hours. When reduced to 3 engines, the flight time increased to 4 hours. Just 2 engines, 6 hours. Only 1 engine, 12 hours. If the pattern continued, what length of time would correspond to 0 engines?”

The joke can serve as a lead-in to inverse variation, and I rather like the answer. With 0 engines, the duration would be infinity. And isn’t that appropriate? If the plane crashes, you won’t reach your final destination for all of eternity!

Speaking of planes, Skyscanner recently conducted a survey about air travel preferences. According to their study, flyers think that 6A is the perfect seat. That shouldn’t be shocking… a perfect seat has to be in Row 6, doesn’t it?

Forty-five percent of respondents said they prefer to sit in the first six rows, and 60% said they prefer the window, so it makes sense that 6A would come out the winner. But there were some surprising results from this survey:

- Nearly 7% said they would choose to sit in the last row. (Really? What kind of person prefers a non-reclining seat by the bathroom?)
- Approximately 62% of respondents said they prefer an even seat number. (Who knew that parity played a role?)
- The worst seat? That distinction belongs to 31E, a middle seat near the back.
- Frequent flyers prefer the left side of the plane. (“Why,” you ask? Because the windows on that side of the plane are off-center, which allows for wall space to rest your head while sleeping.)
- Less than 1% prefer the middle seat. (The surprising part is that
*anyone*prefers the middle seat.)

My guess is that people who prefer the middle seat also think median is better than mode. I assume this result was a statistical error, however. My suspicion is that they asked the preference of two people simultaneously; one responded, “I prefer the aisle seat,” another responded, “I prefer the window seat,” and the survey taker wrote, “On average, these two people prefer the middle seat.”

Recently, I was in seat 6A during an international flight. Halfway home, the entire flight crew got ill from the food. The pilot and co-pilot passed out, so the flight attendants began asking if anyone could fly the plane.

An elderly gentleman, who had flown prop planes over Warsaw many years ago, raised his hand. When he got to the cockpit, he looked at all the displays and controls, and he realized he was in over his head. The flight attendant noticed the look on his face and asked if he was okay. “I don’t think I can fly this aircraft,” he said. “I am just a simple Pole in a complex plane.”

### Jet Setting

What a week! Last Wednesday through Saturday, I took five flights and gave five presentations in four days. The folks at the Denver NCTM Regional Meeting and Northwest Math Conference could not have been more wonderful, and I had a great time, but I’m happy to be home.

On Thursday, I took a connecting flight from Salt Lake City to Spokane that I’d like to forget. Normally a simple, 1.5‑hour journey, this particular flight was shaken by turbulence from take‑off to touchdown, and it didn’t help that I was seated next to a talkative statistician. To boot, about a half‑hour into the air, the pilot announced, “Ladies and gentleman, we’ve lost an engine, but I want to assure you that there’s nothing to worry about. We can still make it safely to Spokane with the other three engines. But instead of just 90 minutes, the flight will now take about 3 hours.”

An inconvenience, sure, but at least we were safe.

A few minutes later, the pilot spoke again. “Folks, it seems that we’ve lost a second engine. We’re still okay, but the trip is now going to take us 6 hours.” The statistician shifted uncomfortably in his seat.

A little while later still, the pilot delivered more bad news. “Ladies and gentleman, I’m really sorry to inform you that we’ve lost a third engine. But I can assure you that we’re still safe. However, the trip will now take 12 hours.”

Upon hearing this, the statistician became agitated. “Good Lord!” he shouted. “I sure hope we don’t lose that fourth engine… or we’ll be up here all day!”

My final talk in Spokane was a math joke hour. After my monologue, I opened the mic to the audience to share any jokes that they knew. The first that I heard wasn’t a math joke, but it’s worth sharing:

What do you call a dog with no legs?

Don’t matter what you call him. He ain’t gonna come.

Another joke pair was visual (read them aloud — the fourth character in each line is a zero, not an O):

IM80D

RU80D2

And finally, the following exclamation was shared:

Holy shift! Look at the asymptote on that mother function!

### Compound Probability

When I told my friend AJ that I had written a book of math jokes, he asked me a question that I found difficult to answer. He asked, “How many will I laugh at?” I paused for a second. Hearing my hesitation, he asked, “Are the jokes really that bad?”

“Well, no,” I explained. “I’m just not sure how many you’ll get.”

AJ is not a dumb guy. He’s quite intelligent, actually. He can hold his own in a conversation with just about anyone on nearly any topic. But some of the jokes in *Math Jokes 4 Mathy Folks* require some advanced understanding of mathematics. Thinking about his question further, I derived the following formula (though not while I was drinking… I never drink and derive):

P(L) = P(G) × P(F|G)

where…

- P(L) is the probability of laughing;
- P(G) is the probability that you get the joke; and,
- P(F|G) is the probability that you’ll think a joke is funny, if you get it.

The question, of course, is how you determine the values for P(G) and P(F|G). Based on absolutely no data whatsoever, I offer the following:

- P(G) = 0.99, if you have a degree in mathematics;
- P(G) = 0.95, if you completed a high school calculus or statistics course;
- P(G) = 0.68, if you completed the minimum high school requirements in mathematics;
- P(G) = 0.51, if you were reasonably successful in mathematics through middle school;
- P(G) = 0.32, if you were okay until your teachers started using words like
*denominator*and*irrational*; - P(G) = 0.03, if you’re a professional athlete;
- P(G) = 0.02, if you’re a member of my extended family (who hate math, aren’t good at it, and are proud to trumpet both facts to anyone willing to listen);
- P(G) = 0.01, if you’re a journalist or other member of the popular media (and possibly lower, if you write for a tabloid).

Of course, I’m egotistical enough to believe that P(F|G) = 1.

So, how many jokes will AJ laugh at? I don’t know. But with over 400 jokes in *Math Jokes 4 Mathy Folks*, there’s got to be a few that he’ll find funny, right?

Anyway, here’s a joke involving compound probability:

When a respected statistician passed through the security check at an airport, a bomb was discovered in his carry-on luggage. “Come with us,” said the security guards, and they took him to a room for interrogation.

“I can explain,” the statistician said. “You see, the probability of a bomb being on a plane is 1/1000. That’s quite high, if you think about it — so high, in fact, that I wouldn’t have any peace of mind on a flight.”

“And what does that have to do with

youcarrying a bomb on board?” asked a guard.“Well, the probability of

onebomb being on my plane is 1/1000, but the chance of there beingtwobombs on my plane is only 1/1,000,000. So if I bring a bomb, the likelihood of there being another bomb on the plane is very, very low.”