Posts tagged ‘temperature’

Two the Hard Way (and an Easy Way)

During our trip to Arizona for winter break, two problems surfaced organically while we were on holiday. (Sorry. Two math problems. There were lots of non-math problems, too, but we don’t have time for all that.)

On the plane, the interactive in-flight map showed the outside temp, toggling between Fahrenheit and Celsius. That led to the following MJ4MF original problem, which I thought — and still think — is pretty good:

The conversion between Fahrenheit and Celsius temperatures follows the rule F = 9/5 C + 32. Sometimes, the temperature is positive for both Celsius and Fahrenheit; sometimes, the temperature is negative for both Celsius and Fahrenheit; and other times, Fahrenheit is positive while Celsius is negative. What is the least possible product of the Fahrenheit temperature and its corresponding Celsius temperature?

You can pause here if you’d like to solve this before I present a spoiler.

Before I reveal two different solutions, allow me to digress. It could be that the statement about the temps sometimes being positive and sometimes being negative is denying a teachable moment. The graph below shows the linear relationship between the two temperature scales. Perhaps a good classroom question is:

When will the product CF be positive and when will it be negative?

Or maybe a better question is:

When are C and F both positive, when are they both negative, and when do they have different signs?

So, to the problem that I posed. As I thought about it on the plane, I concluded that if F = 1.8C + 32, then the product CF = 1.8C2 + 32C. I then used calculus, found the derivative (CF)’ = 3.6C + 32, set that equal to 0, and concluded that C = ‑8.89, approximately. The corresponding Fahrenheit temperature is F = 16, so the minimum product is roughly ‑142.22.

Using calculus was like rolling a pie crust with a steamroller, though. I could have just as easily graphed the parabola and noted its vertex:

If I had used the other form of the rule, namely C = 5/9 (F ‑ 32), things might have been a little easier. Maybe. In that case, setting the derivative equal to 0 yields F = 16, which is arguably a nicer number. But then you still have to find the corresponding Celsius temperature, which is C = ‑8.89, and the product is still roughly ‑142.22. So, not much easier, if at all, and again graphing the parabola and noting its vertex would have done the trick:

The only real benefit to using this alternate version of the rule is that it provides a reasonable check. Since both methods — and both graphs — yield an answer of ‑142.22, we can feel confident in the result.

But there’s an easier way to solve this one.

Thinking this was a good problem — and because I like when my sons make me feel stupid — I gave it to Eli and Alex. Within seconds, Eli said, “Well, F is positive and C is negative between 0°F and 32°F, so the minimum will occur halfway between them at F = 16. That means C = ‑80/9, so it’s whatever ‑1280/9 reduces to.” (Turns out, -1280/9 = ‑142 2/9 ≈ ‑142.22.)

Eli hasn’t taken calculus, so he doesn’t know — or, at least, he hasn’t learned — that the minimum product should occur halfway between the x– and y‑intercepts of the linear graph. Yet, he had an intuitive insight that just happens to be true. As a result, what took me about five minutes of deriving and manipulating took him about five seconds.

The second problem arose at the grocery store. Among our purchases was a box of sugar cubes, which contained, surprisingly, 126 cubes.This number is surprising in the sense that it’s not a number you’ll see very often, except for an occasional appearance in the ninth row of Pascal’s Triangle, or maybe if you’re a chemist searching for stable atoms.

A question that could have arisen from this situation involves surface area and volume:

A rectangular prism with integer dimensions has a volume of 126 cubic units. What is the least possible surface area?

That’s not the question that was shared with Alex and Eli, though. (The answer, if you care, is 162 square units, which results from a 3 × 6 × 7 arrangement — which, in fact, is the exact arrangement of cubes in the box above. I suspect this is not a coincidence.)

The problem that I shared with my sons involved probability:

Imagine that the arrangement of cubes is removed from the box intact, and all six faces of the prism are painted red. Then one of the sugar cubes is selected at random and rolled. What is the probability that the top face of the rolled cube will be red?

The boys made an organized list, as follows:

 Painted Faces Number of Cubes 3 8 2 40 1 58 0 20

Further, the boys reasoned:

• P(cube with 3 red faces, red face lands on top) = 8/126 x 1/2 = 8/252
• P(cube with 2 red faces, red face lands on top) = 40/126 x 1/3 = 40/378
• P(cube with 1 red face, red face lands on top) = 58/126 x 1/6 = 58/756

Therefore,

• (P of getting a red face) = 8/252 + 40/378 + 58/576 = (24 + 80 + 58) / 576 = 162/576 = 9/42

Wow! That seems like a lot of work to get to the answer. Surely there’s an easier way, right?

Indeed, there is.

Notice that the penultimate step yielded the fraction 162/576. The numerator, 162, may look familiar. It’s the answer to the question that wasn’t asked above, the one about the least possible surface area of the prism. That’s no coincidence. In total, there will be 162 faces painted red. And there are 6 × 126 = 576 total faces on all of the sugar cubes (that is, six faces on each cube). This again suggests that the probability of rolling a red face is 162/576.

Did you happen to notice that the volume and surface area use the same digits in a different order? Cool.

So there you have it, two problems, each with two solutions, one easy and one hard. Or as mathematicians might say, one elegant and one common.

It’s typical for problems, especially problems worth solving, to have more than one solution strategy. What’s the trick to finding the elegant solution? Sadly, no such trick exists. Becoming a better problem solver is just like everything else in life; your skills improve with practice and experience. It’s akin to Peter Sagal’s advice in The Incomplete Book of Running, where he says, “You want to be a writer? […] Just sit down and write. The more you write, the better a writer you will become. You want to be a runner? Run when you can and where you can. Increase your mileage gradually, and your body will respond and you’ll find yourself running farther and faster than you ever thought possible.” You want to be a problem solver? Then spend your time solving problems. That’s the only way to increase the likelihood that you’ll occasionally stumble on an easy, elegant solution.

And every once in a while, you may even solve a problem faster than your kids.

Six Degrees of Titillation

It was 75° today. If you live in Honolulu, Karachi, Gaberone, or Rio de Janeiro, that might not strike you as unusual. But let me assure you that during late February in Crofton, MD, a temperature that high is rather unexpected.

But no complaints. It was nice to wear only a t-shirt and no jacket, and it allowed me to use this joke at the start of my presentation:

What a beautiful day! When I was invited to this event, I suspected it’d be close to 20° outside. I just didn’t realize it’d be Celsius.

Horrible, ain’t it? But don’t worry… I’ve got more:

• Do math majors receive degrees or radians?
• If it’s 0° today, and it’s supposed to be twice as cold tomorrow, how cold is it going to be? (Stephen Wright)
• Are you a 45° angle? Because you’re acute-y.
• Why did the obtuse angle go to the beach? Because it was more than 90°.
• Are you cold? Go sit in a corner. It’s 90° over there.
• Why didn’t the circle go to college? It already had 360 degrees.
• I asked the trigonometer what the weather was like, and he said it was 15π/16 outside.
• The number you dialed is imaginary. Please rotate your phone 90° and try again.
• What brand of deodorant did the angle use? Degree.

Fowl Formulae for Thanksgiving

Okay, I know you’re going to find this hard to believe, but there is disagreement on the internet. And I don’t mean about some insignificant topic like gun control or taxes or health care or the value of 6 ÷ 2(1 + 2). This is big. This is important.

We’re talking turkey. Literally.

According to the British Turkey Information Service — yes, there really is such an agency — the amount of time you should cook your turkey at 375° F can be found with the following formula:

$t = \begin{cases} 20w + 70 & w < 4; \\ 20w + 90 & w \geq 4, \end{cases}$

where t is the cooking time in minutes and w is the weight of the turkey in kilograms.

If you’d rather not do the math yourself, try the British Turkey Cooking Calculator, which will not only give you the cooking time but also the defrosting time and the size of turkey to buy for a given number of servings.

By comparison, the Meat Chart provided by FoodSafety.org says that turkey should be cooked at 325° F for 30 minutes per pound.

But the cooking website allrecipes.com says that only 20 minutes per pound is sufficient if you bake the bird at 350° F.

Whereas the good folks at delish offer the following guidelines:

Cooking Times at 325° F from delish.com

which translates to the lovely formula

$t = \begin{cases} 5w + 125 & w \leq 10; \\ 15w & w = 12; \\ 7.5w + 120 & w \geq 14, \end{cases}$

but requires that you interpolate if your bird weighs an odd number of pounds. (Like 86 pounds, the world record for heaviest turkey ever raised. Even though the units digit is 6, you’d agree that 86 is an odd number of pounds for the weight of a turkey, no?)

As you might suspect, Wolfram Alpha has a more mathematically sophisticated formula:

$\displaystyle t = T \times \left( \frac{w}{20} \right)^\frac{2}{3}$

where t is the cooking time in hours, w is the weight in pounds, and T is a coefficient to account for cooking environment. For normal conditions, T = 4.5, and the equation reduces to

$\displaystyle t = 36.64w^\frac{2}{3}$

if you use minutes instead of hours for the unit of time.

But this feels a little like a math joke; below the formula, Wolfram offers the following:

using the heat equation for a spherical turkey in a 325° F oven

Falling into the wrong hands, that idea could lead to an horrendous modification of the spherical cow joke…

The turkeys at a farm were not gaining sufficient weight in the weeks leading up to Thanksgiving, so the farmer approached a local university to ask for help. A theoretical physicist was intrigued by the problem and offered his assistance. He spent several weeks at the farm, examining the turkeys and filling his notebook with equation after equation. Finally, he approached the farmer and said, “I have found a solution.”

“Oh, that’s excellent!” said the farmer.

“Yes,” said the physicist. “Unfortunately, it only works for spherical turkeys in a vacuum.”

The Wolfram formula is very similar to one suggested by physicist Pief Palofsky, who apparently dabbled in poultry when not winning the National Medal of Science.

$\displaystyle t = \frac{2}{3} w^\frac{2}{3}$

and when converted to minutes instead of hours, this becomes

$\displaystyle t = 40w^\frac{2}{3}.$

According to Turkey for the Holidays, the average weight of a turkey purchased at Thanksgiving is 15 pounds. The cooking times for a 15-pound bird, based on the formulae above, appear to have been chosen by a random number generator.

 Recommender Time (min) Temp (° F) British Turkey Information Service 226 minutes 375 Foodsafety.org 450 minutes 325 allrecipes.com 300 minutes 350 delish.com 233 minutes 325 Wolfram Alpha 223 minutes 325 Pief Palofsky 243 minutes 325

Even if you limit consideration to those who suggest a cooking temp of 325° F, the range of times still varies from just under 2¾ hours to a staggering 7½ hours. Wow.

With Thanksgiving just around the corner, where does all of this contradictory information leave us?

A number of sites on the internet claim that the only way to adequately check the doneness of a turkey is with a meat thermometer.

The folks at recipetips.com claim that a turkey can be removed when the temperature is at least 170° F for the breast and 180° F for the thigh. Yet on the very same page, they claim, “Turkey must reach an internal temperature of 185° F.”

On the other hand, the folks at the Food Lab claim a turkey can be safely removed when the breast temperature reaches 150° F, because after resting 15‑20 minutes before carving, the amount of remaining bacteria will be minimal. They explain, “What the USDA is really looking for is a 7.0 log10 relative reduction in bacteria,” particularly Salmonella, which means that only 1 out of every 10,000,000 bacteria that were on the turkey to start with will survive the cooking process. And according to the USDA guidelines, a turkey that maintains a temperature above 150° F for 3.8 minutes or longer will reach that threshold for safety.

Which has to make you wonder — if 3.8 minutes at 150° F is supposedly adequate, why then does the USDA Food Safety and Inspection Service recommend that the minimum internal temperature of the turkey in the thigh, wing, and breast should be at least 165° F? Who knows. I suspect it’s typical government over-engineering to remove all doubt.

So, how long should you cook your turkey? Hard to say. But if you put your turkey in the oven right now, it should be done by November 23.

When the turkey is finally ready, here are a few math jokes you can tell around the Thanksgiving table.

What do math teachers do on Thanksgiving?
Count their blessings!

What does a math teacher serve for dessert on Thanksgiving?
Pumpkin Pi.

How do you keep private messages secure on Thanksgiving?
Public turkey cryptography.

Thanksgiving dinners take 18 hours to prepare. They are consumed in 12 minutes. Halftimes take 12 minutes. This is not coincidence.
~ Erma Bombeck

Gobble, gobble!

Math in France

Driving through the French countryside using smartphone GPS for navigation is a lot like driving through rural Pennsylvania with my redneck cousin riding shotgun — there is a significant lack of sophistication, an ample amount of mispronunciation, and myriad grammatical errors.

In Pennsylvania:

Take that there right onto See-Quo-Eye-Ay (Sequoia) Drive.

In France:

At the roundabout, take the second right toward Ow-Bag-Nee (Aubagne).

Take D51 to Mar-Sigh-Less (Marseilles).

And of course, the GPS pronounced the coastal town of Nice like the adjective you’d use to describe your grandmother’s sweater, though it should sound more like the term you’d use to describe your brother’s daughter.

I was half expecting the computer voice to exclaim,

Hey, cuz, watch this!

Otherwise, the rest of my recent week-long trip to the south of France was intellectually and often mathematically stimulating. The image below shows a -1 used to describe an underground floor (parking) in a hotel:

And though I didn’t get a picture, the retail floor of the parking garage at the Palais de Papes in Avignon was labeled 0, with the three floors below for parking labeled -1, -2, and -3.

This is a country that does not fear negative integers.

I also noticed that the nuts on fire hydrants in Aix-en-Provence were squares.

Hydrant with Square Nut

The nuts on American hydrants used to be squares, until hoodlums realized that two pieces of strong wood could be used to remove them, release water into the streets, and create an impromptu pool party for the neighborhood. As a result, pentagonal nuts are now used on most hydrants.

Alas, an adept hoodlum can even remove pentagonal nuts, so some localities have replaced them with Reuleaux triangle nuts, like the ones on hydrants outside the Philadelphia convention center, which can only be removed with a specially forged wrench.

with Reuleaux Triangle Nut

But perhaps the most mathematical fun that France has to offer is the Celsius scale. While there, our cousins taught my sons a poem for intuitively understanding the Celsius scale:

30 is hot,
20 is nice,
10 is cold,
and 0 is ice.

And I was able to teach them a formula for estimating Fahrenheit temperatures, which is easy to calculate and provides a reasonable approximation:

Double the (Celsius) temperature, then add 30.

Or algebraically,

F = 2C + 30

The actual rule for converting from Fahrenheit to Celsius is more familiar to most students:

F = 1.8C + 32

This rule, however, sucks. It’s not easy to mentally multiply by 1.8.

My sons were not convinced that the rule for estimating would give a close enough approximation. I showed them a table of values from Excel:

I also showed them a graph with the lines y = 1.8x + 32 and y = 2x + 30:

With both representations, it’s fairly clear that the estimate is reasonably close to the actual. For the normal range of values that humans experience, the estimate is typically within 5°. Even for the most extreme conditions — the coldest recorded temperature on Earth was -89°C in Antartica, and the hottest recorded temperature was 54°C in Death Valley, CA — the Fahrenheit estimates are only off by 9° and 20°, respectively. That’s good enough for government work.

And here’s a puzzle problem for an Algebra classroom, using this information.

The Fahrenheit and Celsius scales are related by the formula F = 1.8C + 32. But a reasonable estimate of the Fahrenheit temperature can be found by doubling the Celsius temperature and adding 30. For what Celsius temperature in degrees will the actual Fahrenheit temperature equal the estimated Fahrenheit temperature?

It’s not a terribly hard problem… especially if you look at the table of values above.

If This is What Hades is Like, We Better Change Our Ways

It was so hot in Virginia today, I saw two fire hydrants fighting over a dog.

Truth be known, I’m travelling through California right now, so I don’t actually know how hot it was in Virginia today. But I got an email that said my wine o’ the month shipment would be delayed until the weather cools to a point where the wine won’t be in danger of spoiling. Zoiks! In California this week, it’s been rather pleasant — even chilly at night, with temps in the low 50’s.

A recent conversation with the Northern California friends we’re visiting turned to temperature conversions. John said that on international trips, he amuses himself by converting Celsius temps to Fahrenheit. I informed him that the rule “times 2 plus 30” gives an accurate estimate for temps in the typical range — it isn’t really necessary to remember the exact rule. I also shared a poem about Celsius temps that he and his wife hadn’t heard before:

30 is hot,
20 is nice,
10 is cold,
and 0 is ice.

Finally, a stupid math and weather joke.

What is the name of a person who only adds when it’s hot outside?
Summer.

The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.

MJ4MF (offline version)

Math Jokes 4 Mathy Folks is available from Amazon, Borders, Barnes & Noble, NCTM, Robert D. Reed Publishers, and other purveyors of exceptional literature.