Sometimes this stuff is just *sitting there* for the taking, like a gold brick in the middle of the road, and all it takes is the right kind of eye to see it. David Petro e-mailed me a link to an episode of the long-running radio series, *Car Talk*, which does all the heavy lifting for us. I just shot a couple of supplementary photos.

**Start with discussion, brainstorming, estimation.**

Rich, *Car Talk* caller [show page, direct download]:

The fuel gauge [on my 18-wheeler] is completely useless. It’s completely unreliable. So I have gone with a very low-tech method. I have gotten a wooden dowel which I use as a dipstick. Now the tanks on my 18-wheeler are cylinder-shaped and they’re on their sides. Now when I stick the dipstick in and I mark the level when the tank is completely full, it happens to be 20 inches. I mark ten inches at the half.

My question is how I do I accurately mark one quarter?

I mean, *seriously*? Christmas *already*?!

I’d crop this clip pretty tightly. Rich says, “It obviously isn’t five inches,” but we’d rather put that to our students, asking them, “**Just gimme a guess**: how many inches do you think?” Some students – the impatient ones, maybe even a few advanced students – will snap to “five inches.” **Have them sketch out their solution.** Ask them if they still like it. Ask for a number they know is too large, too small.

**Important questions your students may have to reckon with.**

- What information do we care about here? What information
*don’t*we care about here? - Why does the trucker care about the level of a quarter tank anyway?
- If you try out your guess, is there more than or less than or exactly a quarter of a tank left?
- What kind of shape does the gas form in the bottom of the tank?
- How is that shape formed from other shapes and how could we find its area?

These questions and their answers will vary significantly with the particular math course you’re teaching.

**Answer**

One of the show hosts: [show page, direct download]:

We finally came up with the answer, that in order for his dipstick to work he’s gotta be about 5.96 inches off the bottom. That’s a quarter of a tank. And it seemed like it should’ve been a lot more than that.

**The Goods**

Voila: a zipped archive. Thanks, David.

**Extra Credit Assignment**

Someone cook up a dynamic Geogebra applet for this scenario and pass me a link.

**BTW**: Woo! Check out Tim’s.

**BTW**: Updated to fix an error in my math. Thanks, **RM**.

## 27 Comments

## paul

December 8, 2010 - 4:22 pm -how does this work for you

http://dl.dropbox.com/u/15353231/gastankfinal.ggb

i need to use this concept in geometry class

## Dan Meyer

December 8, 2010 - 4:29 pm -You’re an animal.

I was picturing a slider for x, the depth of the gas in the tank, with an updating counter for its percentage of the entire tank. What do you have for me?

## paul

December 8, 2010 - 4:53 pm -i have not played with sliders…hopefully someone else can add that.

## Tim

December 8, 2010 - 5:31 pm -Here’s my question: When he measured 10 inches, how did he know that it was exactly half full?

## Annie

December 8, 2010 - 5:47 pm -@Tim – he didn’t have to. If the cylinder is actually full and the diameter = 20″, then half full = radius = 10″. Based on the transcription above, I read it as though he only measured the 20″, not the 10.

This is brilliant, btw. Sharing with everyone I know.

## Tim

December 8, 2010 - 5:59 pm -Dan, http://math.temple.edu/~tub97278/two_7.html

Feel free to change the extension to .ggb to get the source.

## Dan Meyer

December 8, 2010 - 6:05 pm -@

Timwoo! That’s hot.## paul

December 8, 2010 - 6:07 pm -Tim, that is nice, much less clutter than my attempt.

Teaching Geometry I would still like them to see getting the area of a segment by subtracting the triangle from the sector.

None of my sophomores has had sin/cos/tan yet, though they should have (we will hit it soon)

## Chris Sears

December 8, 2010 - 6:12 pm -I was just talking about this in my Calculus II class, only with a twelve foot diameter milk truck. They had their final today, so I will have to share this with them next semester.

## Barry McMullin

December 9, 2010 - 2:36 am -And just in case you need a follow on challenge for the really

bright kids: for my home heating I have (or used to have) a

cylindrical oil tank; and I use a dip stick to check it, just

like the trucker. Except the oil tank is not on the level – it

is sloped along the axis of the cylinder so that one end is a

little higher than the other. This is not accidental: it’s to

ensure that the oil flows down to the outlet pipe even when it is

getting low. From a practical use point of view this is a

natural and very minor variation; but the relationship between

dipstick height and volume is now a good deal more

complicated. Yet … it’s exactly when the oil is getting low

(i.e., when I’m most interested in accuracy) that the discrepancy

from the level tank case is greatest. I guess … So WCYDWT!

(Confession – this is an “authentic” problem: About 30 years ago

when I first moved into my “own” home, I had exactly this

situation; and by golly I decided I was going to do the maths and

solve it. I probably still have those scraps of paper lying

around somewhere …)

## Daniel Schaben

December 9, 2010 - 6:09 am -Here is one I did with my father after hearing the car talk

episode.

http://blog.esu11.org/dschaben/2010/11/22/how-much-fuel-is-in-the-truck/

Sorry for the shameless plug. Cool problem

## Karl

December 9, 2010 - 6:35 am -Doesn’t it matter that the gas port is not on the top of the tank?

It would need to be offset at some angle from vertical, so you are not putting your dipstick through the center of the circle, but cutting across. (I forget my geometry terminology, having switched to physics almost ten years ago… is it called a cord?)

Seems to me that makes the problem both more interesting and (naturally) more difficult…

## Benji

December 9, 2010 - 7:47 am -Karl: so long the rod he is using is pushed to its furthest extent, it would be covering the longest distance of the circle, that being the diameter. Since the gas port is cylindrical, I am assuming when he puts it in, his force would slide the rod to its furthest position.

I agree with your idea of using a chord to solve this problem, though. Maybe the hole for the gas port is too small for the angle of the rod to hit maximum distance?

## Karl

December 9, 2010 - 10:11 am -Maybe I didn’t quite describe it right Benji. The problem is that the gas has to be at the bottom of the tank, so unless the gas port is right on top, the diameter that passes through the gas port will not actually touch the gas at all when the tank is low.

A diagram would help but I don’t know how to put one in comments…

In the diagram above, picture if the gas port is about where the arrow sticks out of the circle on the top right. Even if your dipstick hits the bottom of the tank, it will intercept the surface of the fuel air interface at a non right angle…

Does that make sense?

## Benji

December 9, 2010 - 11:44 am -What I think you’re saying (and I think I’ve got it now :D) is that when you insert the “gas tank measurement rod” that if the access hole is positioned off from directly above then when penetrating, it might not measure the correct amount of gas.

In your theory, if the gas tank is low enough and the angle is just right, you could pull out a dry measuring rod BUT still have gas in the tank (like the picture you referred to)?

Interesting hypothesis… should we create a new instrument that measures straight down? or re-evaluate our problem??

## Daniel Schaben

December 9, 2010 - 11:45 am -Tim – Could you make your green line shift left or right? In the problem that I produced the entry hole for the tank is not at the top of the tank. It is slightly left or right of center depending on your orientation. When my father dipped the rod in and stayed parallel to the center line it gives a reading of 6.5 inches. If we could have dipped from the very top of the tank the measurement is more than likely 7 inches. I am going to have to go back to the drawing board on this problem and show some more measurements. Great post. This is possibly my favorite problem in geometry because it seems so easy when you first look . . . but then BLAM trigonometry. pi*r^2*h is no longer enough.

## RM

December 9, 2010 - 6:10 pm -Should the attached drawing have (1/4)pi(10)^2 instead of (1/4)pi(20)^2?

I set up a similar equation. How do we solve for x?

## Ben Blum-Smith

December 10, 2010 - 5:13 am -Scooped! I had this filed and I was totally going to post on it! (But I committed to not posting again until a) my grad school applications are in and b) I was finished with a particular long-overdue post that’s not this. So I got rolled on ;)

But this is a much better outcome. My post wouldn’t have reached the same readership nor had the same dramatic photos.

## Sue VanHattum

December 10, 2010 - 8:40 am -Barry, I

loveyour problem. It sounds much harder, though. Do you have any details? I might offer it as extra credit somehow to my calc II students next semester.## Dan Meyer

December 10, 2010 - 10:13 am -BarryandKarl’sextensions are fab.## Mike Manganello

December 11, 2010 - 11:26 am -I’m a little uncomfortable with the definition of pseudocontext being used, as well as what differentiates a WCYDWT problem and an exercise laden with pseudocontext. I can certainly accept working definitions that require clarification, but the Car Talk problem confuses the issue (at least for me). I’ve only done a little tweaking to the Car Talk problem:

The fuel gauge of an 18-wheeler is broken, so the driver decides to check the gas level of his cylindrical gas tank with a dipstick. When the level of the gas measures 20 inches high, the tank is completely full. What will the dipstick measurement be when the gas tank is one-quarter full?

Based on the working definition of pseudocontext, this problem fails on both counts. It completely ignores reality: Why wouldn’t you just fix the gas gauge? Then the problem asks for an irrelevant measurement: Why would we need to know that the tank is one-quarter full? (Personally, I find the Car Talk problem kind of boring and not very mathematically rich.) The only thing that rescues this from the insurmountable depths of end-of-chapter textbook exercise is that it was generated by a real person. Is what divides a WCYDWT problem from a pseudocontext problem is that an actual person generates it from a genuine need? Is it an issue that it is so easy to transform a genuine WCYDWT problem into a textbook-like exercise? While it is certainly true that there are pseudocontext problems that would (probably) not manifest themselves in someone’s actual existence, there are plenty of exercises that were originally born from a genuine situation and have found their way into mass-produced textbooks. Does that make them irrelevant? Or do we need a better filter?

Another word of caution: Mathematics is part utility and part artistry. By limiting mathematical study to problems related to genuine physical phenomena can only serve to retard the growth of mathematics. Many branches of mathematics, especially those pioneered last century, could not survive if physical use for the mathematics were a necessary condition of research. (For example, the cubic formula would never have been developed if Tartaglia and Cardano had not manipulated of square roots of negative numbers without really understanding what they were.) Indeed, sometimes mathematicians describe patterns, not because the patterns are useful but because the patterns are there.

## James McKee

December 13, 2010 - 7:42 am -I actually had a trucker come to me a couple of years ago with this problem. He wanted a dipstick marked off so that he could stick it in the tank and know how many gallons of fuel he had on board.

I went a little different direction and created a spreadsheet that gives the gallons for any particular depth (in inches, since he’s old-school), so that he could use any yardstick/meterstick, or for that matter a tape measure, and we didn’t have to worry about deterioration of the stick.

However, it strikes me that creating a dipstick that is marked in regular gallon intervals (say every 10 gallons) would be an interesting question. What would such a dipstick look like, and how could we go about creating it?

## Alex

December 13, 2010 - 1:02 pm -IMHO one of the best WCYDWT problems I’ve seen.

Of course my kids said, “Why not just fix the gas gauge?” But that was easily dealt with by saying that if he’s run out of gas twice and had to pay to get towed or have some trucker version of AAA bring him 90 gallons of diesel that maybe he’s strapped for the cash. And it sure didn’t stop them from delving into the problem with sincere fury. It was incredible to watch.

For my geometry class it was an awesome opportunity to use multiple concepts at once:

Trig functions

Area of a circle

Area of a sector of a circle

Area of a triangle

Very well done. Keep picking up those bricks of gold :)

## paul

December 14, 2010 - 6:17 pm -just found this

http://blog.virtualnerd.com/2010/11/filling-a-tank-to-a-quarter/

## Monica

December 15, 2010 - 6:23 am -I don’t know much about 18 wheelers, but I do know what I did when my car’s gas gauge was broken…I used the odometer and my car’s average mi/gal. To this day, the odometer gives me a better sense of how far I can drive than my gas gauge.

This is a different problem, but a realistic one that students are more likely to experience in their lives. I don’t know of any cars that have their fuel tank exposed or a cylindrically shaped tank.