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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

    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 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)

    Oops! A GOOF! The answer is SIX!!

    by cmoverdahl@aol.com

    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 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.

    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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