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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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Use string to solve the cylindrical gas problem

by Pfleming91

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.

Show Review - 1376

by peterirlenborn

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.

Excel to the Rescue

by RuKaz

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

MIT ..... Ha!

by nyandmaine

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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