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#1045: Pi Over Two Dopes

Nov 06, 2010
This week on Car Talk, trucker Richard has a busted fuel gauge in his 18-wheeler. He's got a trusty stick to help measure the level of fuel in his cylindrical tank, but needs a formula to tell him whether that level means he's got 3/4 of a tank, or is running on fumes. You'd think two MIT grads would be able to help, but, well, you'd be wrong. Happily, Tom and Ray are more on the ball in recommending a snow-worthy vehicle for Gail, her husband, and their 8 kids, and in helping Louann figure out why her Accord smells like burnt toast. Also, why "oil replenishment" is like vacuuming your living room without a bag, and Tommy goes on the rampage against--Gomer Pyle? All this and lots more, this week on Car Talk.

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Kids say the funniest things... and Tom and Ray have the examples to prove it!

This Week's Puzzler

Can the sales-kid explain where the dollar missing from the record store register went? Can you?

Last Week's Puzzler

Dave pronounces his nephew's new atomic watch "stupid" for a particular reason. But Dave's nephew proves him wrong. What was Dave wrong about?

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21 Comments

Use string to solve the cylindrical gas problem

As an alternative to the center of mass solution (which some sticklers have rejected) I would offer this method to the trucker. It has the advantage of not requiring any numerical work or calculus. I'm going the assume that the tank has unit radius. The solution can always be scaled by your favorite units. As others have noted, the area between the diameter and a chord parallel to the diameter is A + sin(2A)/2 where A is the (acute) angle that the diameter makes with a radius from the center of the circle to an endpoint of the chord. (This is really a challenge without pictures!) The trucker would like to find A such that pi/4 = A + sin(2A)/2. An easy way for him to do that which avoids messy calculation is to observe that the above equation is equivalent to pi/2 = 2A + sin(2A) and if he can determine 2A then he can easily find A. To find 2A: He finds a circular piece of cardboard (e.g., from a pizza box) of the same diameter as his gas tank and a piece of string whose length is one quarter the circumference of the cardboard circle. Starting at one end of a diameter he carefully wraps the part of the string around the cardboard circle up to a point (call it P) on the circumference so that the remainder of the string will drop down perpendicular to the diameter and just meet it. The straight part which is perpendicular to the diameter will then have length sin(2A) and the part wrapped around the edge of the cardboard circle will have length 2A. The trucker now can use the string to find A from 2A, draw the desired chord parallel to the diameter and measure the distance from the chord to the diameter and subtract it from 1. I hope this is enough of a description for the reader to draw a picture of the solution.
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Show Review - 1376

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You are trying tooooo hard!!

It is easier to figure the 1/4 of a tank than all your MIT methods. He said he knew how much the tank held and where the 1/2 mark is. Say the tank holds 100 gallons. Half would be 50 gallons. You have the half mark. Fill to that mark. Now add 25 gallons and mark where the line is for 3/4 full. The distance on the line from 3/4 to the top is 1/4. That line is also the distance from the bottom UP that will mark 1/4 of the tank.
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I love to hear the guys laugh. It is infectious!!!!!
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simple solutions

I love the show I don't drive so I obviously don't have a car but, I thought you would enjoy this quote from a MacGyver Christmas ad that I recalled after hearing the beer can solution to the truck drivers problem "Sometimes the hardest looking problems are solved by the simplest solutions" your show always brings this quote to mind
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10am to 11am on VPR and 11am 12pm on WBUR
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1/4 full at 6 inches on stick

You had a great opportunity to show off your calculus skills! I used a spreadsheet to estimate the dipstick level and then checked it with a google.com search for "horizontal tank volume" The spreadsheet took an hour and the google search took 5 minutes! The fuel dip stick level for 1/4 tank remaining if the tank depth is 20 inches is 6 inches. (See http://www.1728.com/cyltank.htm)
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teach your girl to drive stick

I usually love your show, but as a former 16-year old girl (roughly 16 years later) I took offense at the recommendation to teach a teenage girl to drive an automatic first, and a stickshift later. Thanks to my Dad's patience and my parents only having stickshift cars, I begrudgingly learned how to drive a manual transmission while all my friends had instant success in automatic cars. I have to say it was the best decision my parents made; since then I have been able to drive anything, from a 1969 VW bug in college to a fiesty pick-up truck for an on-campus recycling crew that only went into 3 gears. Teach her a stick and a stick only now - any idiot can drive an automatic. If we assume our teenage girls are so distracted by their cell phones and getting used to the steering wheel they will sink to that expectation. Teach them to handle a stick, check and change their own oil, and change a flat. They will feel confident for the rest of their lives, and can always fall back on an automatic and AAA later.
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Solving the Cylinder Problem

My brother told me about the problem you guys were trying to figure out with the truck driver's gas tank. I'm a math teacher and I tried my best to figure it out. If the tank measured 20in across in diameter, the quarter tank mark, (and 3/4 tank mark), should be 4 inches above or below the middle of the stick used to measure the diameter. I solved it by taking a formula for the area of the quarter of a circle, and set it equal to the formula of a circle's segment: (pi*r^2)/4 = ((A - SinA)r^2)/2 Then I solved for A, which is the central angle for the segment, and then I used the central angle to find the length. I hope this is helpful.
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Excel to the Rescue

Please see submission written before I registered. Your instructions said "if either is missing," but I was missing only the password, so went ahead.
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The answer: 4 inches!

Your e-mail reviews have 2 answers so far: 6 inches, and 4 inches. The math teacher is correct that (about) 4 inches in depth equals one-quarter tank. By the way, the solution uses Trigonometry, and not Calculus (as mentioned on the show). So much for MIT.
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Try Excel Solver

My husband and I are an engineer and a rusty mathematician, but we independently came up with roughly the same answer (mine was the more precise 5.9602724...) for the distance from the bottom of the barrel to the fuel level, such that the area of the cross-section of the barrel content is 1/4 that of the cross-section of the full barrel. The length of the cylinder is irrelevant, so we can work with area instead of volume. My husband derived his own equations, and used trial and error to obtain the approximate answer of 5.96, while I used Excel Solver add-in to get the precise answer above. We each ended up with one trigonometric/transcendental equation, and several algebraic ones -- NO calculus!
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Oops! A GOOF! The answer is SIX!!

I looked over my scratch paper, and saw an error! (Translation for MIT grads: a GOOF). The answer is, in fact, (about) 6 inches. (4 inches is what you calculate, but then it needs to be subtracted from the radius of 10).
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Horizontal Cylinder volume calc

I'm certain you guys had this (exact same) problem once before. I don't recall whether it was a puzzler or a call-in question, but I do remember that a professor (maybe MechE) sent you a detailed derivation of the formula, including charts and graphs.
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MIT ..... Ha!

Oh you guys really did a disservice to MIT on this one. You couldn't figure out a simple trigonometric relationship, and went so far as to claim you needed calculus for it! Shame on you. For trucker Richard, the volume of a cylindrical tank is easy. (That is ignoring the rounded end caps.) For a cylinder of radius, R, and the height of the dipstick, X, we get: %Fuel = ((R**2)*arccos((R-X)/R)-((R-X)**2)*tan(arccos((R-X)/R))/PI*R**2 Put into a spreadsheet, we get for a 20" diameter tank: Depth %Fuel 0 0.000 1 0.019 2 0.052 3 0.094 4 0.142 5 0.196 6 0.252 7 0.312 8 0.374 9 0.436 10 0.500 11 0.564 12 0.626 13 0.688 14 0.748 15 0.804 16 0.858 17 0.906 18 0.948 19 0.981 20 1.000 This says at 6" he has a quarter tank left. Where you really blew it was your comment about teaching a person to drive a manual or automatic. I strongly disagree with your conclusion. Learning on a manual gives you the ability to drive anything in the world. The initial learning process is better, as learning on the manual inherently causes the driver to pay more attention to the car, and what it is doing, rather than being distracted with other incidentals. Typically, when a young person learns on an automatic, it is very likely she will NOT want to ever go to the more complicated manual, and will likely refuse all future attempts at doing so. On another note, I take exception to your disparaging remarks about Brooklyn. Being a former Brooklynite, we "speak real good." It is the rest of the world that doesn't get it right. Especially for a couple of guys from Cambridge, you don't even know how to "park a car."
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Then there's the stupid approach

If you don't feel like looking up the formula for the area of a section of a circle, or doing trigonometry or integrals to derive it, there are much easier ways to get the answer. You can draw a circle on graph paper, count up squares, and get an answer close enough for any trucker's purposes. Or you can use an experimental model: get a can of soup, punch a hole in the side to emulate the gas tank fill hole. Empty out the soup (have it for lunch), fill the can with water, and empty it into a measuring cup to determine the volume (I'm not assuming here the soup filled the volume completely). Now put 1/4 of the water back, stick a toothpick or something in to act as a dipstick, and measure the height of the water. Scale up to the radius of the gas tank and you're all set, not a soh-cah-toa in sight.
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gas tank problem

In order to figure out the 1/4 and 3/4 full heights of the gas tank, it is useful to consider only the circular cross section of the tank and first plot an estimate of what the volume (or area in this simplified case) of the tank would look like with height of the liquid in the tank. Once you have this plot, you can see that it resembles an inverted cosine function. Using the rules of adjusting cosine functions to fit values at certain points, the resulting function is: -((Pi*r^2)/2)cos(h*Pi/2*r)+(Pi*r^2)/2. By setting this function equal to (Pi*r^2)/2, you can solve for h and check that you get r. Once you are satisfied that the equation is correct, you can set it equal to (Pi*r^2)/4 (or 1/4th the total volume) and again solve for h, which gives a solution of 9.2 inches.
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Fuel Tank Solution

If you want to use calculus, here is the solution: Come up with a 2-D section of a circle: x^2 + y^2 = r^2, or x^2 + y^2 = 100. Set y = sqrt(100-x^2). Integrate from t to 10, t being the distance from the center of the fuel tank down to the surface of the fuel at 1/4 full. Set this integral equal to 1/8 the the Area of the full circle (it's one-eight since you're only calculating for positive values of the y-axis). 1/8 of the area is 39.27 in^2. t comes out to be about 4.04. Thus, the distance from the top of the fuel tank is 14.04 inches. (Note, integrating sqrt(100-x^2) is difficult and I had to use an online integrator calculator)
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14 inches...

The necessary calculation can be found from http://mathworld.wolfram.com/CircularSegment.html mid page it says: To find the value of h such that the circular segment has area equal to 1/4 that of the circle, plug A=pi*R^2/4 into equation (18) and divide both sides by R^2 to get (23) where x=h/R. This cannot be solved analytically, but the solution can be found numerically to be approximately h=0.596027 or r=0.403973 So to measure a quarter tank, mark your dipstick at 14 inches from the top or 6 inches from the bottom.
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14 inches...

The necessary calculation can be found from http://mathworld.wolfram.com/CircularSegment.html mid page it says: To find the value of h such that the circular segment has area equal to 1/4 that of the circle, plug A=pi*R^2/4 into equation (18) and divide both sides by R^2 to get (23) where x=h/R. This cannot be solved analytically, but the solution can be found numerically to be approximately h=0.596027 or r=0.403973 So to measure a quarter tank, mark your dipstick at 14 inches from the top or 6 inches from the bottom.
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What's to review?

I'd love to give a good review of this show but I can't. Because I can't hear it. I listen online and I clicked on all the links I usually click on and the pop-up window appears but no sound. (Yes, I checked my speakers.) What gives?
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I'm still waiting for it.
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GaaaaLeeee

I take offense to your views of Gomer Pyle! What American hates Gomer Pyle???
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Math problem

As noted by several, don't need calculus and can be solved in 2D. A drawing and simple trig gives the area of a sector of a circle of radius r and distance h below the center as r^2(t-0.5 sin (2t)) where t=acos(h/r). Set this equal to Pi r^2/4 (quarter area of circle) and solve for t. This can be done numerically using Newton's method, or if you dont like to use calculus at all, using simple iteration, t(n+1)=Pi/4+0.5 sin (2 t_n), which converges to t=1.1549 rads, from which h/r can be computed. For a dia of 20 inches, h=4.04 measured from the center. Hence the quarter tank mark will be at 10-4.04 or 5.96 inches.
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