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#1045: Pi Over Two Dopes

Original Air Date: 11.06.2010

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Description: 
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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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."

What's to review?

by Amy

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?

Favorite Moment: I'm still waiting for it.

Then there's the stupid approach

by doctroid

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.

Try Excel Solver

by RuKaz

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!

gas tank problem

by kpederso

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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