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#1045: Pi Over Two Dopes

Original Air Date: 11.06.2010

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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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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? Find out!

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

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Solving the Cylinder Problem

by tads

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.

The answer: 4 inches!

by Anonymous

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.

teach your girl to drive stick

by eborshard

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.

1/4 full at 6 inches on stick

by MikeVV

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 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

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).

14 inches...

by Anonymous

The necessary calculation can be found from 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.

Math problem

by arun112977

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.

You are trying tooooo hard!!

by Anonymous

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.

Favorite Moment: I love to hear the guys laugh. It is infectious!!!!!

simple solutions

by Anonymous

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

Favorite Moment: 10am to 11am on VPR and 11am 12pm on WBUR

Horizontal Cylinder volume calc

by SRovinsky

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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