1/4 Tank Dipstick Problem (from Car Talk)
Date: 12/04/2002 at 13:30:36 From: HydroJ Subject: Dipstick problem... This past week on the National Public Ratio radio program Car Talk, Tom and Ray Magliozzi got a call from Rich from Florida, who was actually on the road in Missouri. Here's how they state his intriguing mathematical conundrum: The gauge on his 18-wheeler is kaput. So he uses a dowel to measure the diesel in his tank - which just so happens to be cylinder-shaped and 20 inches in diameter, and which sits on its side. Now, any moron (that would be us) could tell Rich that his tank was half empty when the stick measured 10 inches of fuel. But when is the tank exactly three-quarters empty? Or a quarter empty? We tried, but got lost in the calculus. Thanks... - Hydro
Date: 12/10/2002 at 23:18:49 From: Doctor Ian Subject: Re: Dipstick problem... Hi Hydro, Looking at the tank from the side, we see a circle: * * | * * | h | A.................B | * C * * * * * * * What we'd like to do is find h such that the area above the chord AB is some fraction of the total area, so that we can make an appropriate mark on the dipstick. Our FAQ on segments of circles, http://mathforum.org/dr.math/faq/faq.circle.segment.html explains how to take any two of the standard attributes of a segment and use them to find all the others. Case 13 tells how to proceed when we know the radius (r) of the circle and the central angle (theta) the subtends the segment. Of course we don't know theta yet, but it will be useful to pretend for a moment that we do. (In the illustration above, theta is the angle ACB.) The area of the segment is given by theta - sin(theta) r^2 ------------------ 2 In the general case, we would like for the area of the segment to be some fraction a/b of the area of the entire circle. For illustrative purposes, let's set a/b equal to 1/4. Now we can set the area of the segment equal to 1/4 the area of the entire circle: theta - sin(theta) pi r^2 r^2 ------------------ = ------ 2 4 We can cancel r^2 from both sides to get theta - sin(theta) pi ------------------ = ---- 2 4 and multiply both sides by 2 to get pi theta - sin(theta) = ---- (approximately 1.57) 2 Note that pi/2 radians is 90 degrees, so theta is going to be larger than 90 degrees. How much larger? Could it be 120 degrees? That gives us 2 pi/3 - sin(2 pi/3) = 2.09 - 0.87 = 1.22 which is a little low. How about 150 degrees? That gives us 5 pi/6 - sin(5 pi/6) = 2.62 - 0.5 = 2.12 which is a little high. So it's somewhere between 120 and 150 degrees, and you can narrow this down as precisely as you want by guessing. (If you're in a hurry, you can use the numerical method described at the bottom of the FAQ on segments of circles to make more efficient guesses.) A little fooling around with a spreadsheet says that 132.3 degrees is pretty close to the actual value. In any case, suppose we find the value of theta that makes the area of the segment equal to 1/4 (or any other fraction) of the area of the circle. Then what? Back to Case 13: h = r - r cos(theta/2) = r(1 - cos(theta/2)) = r(1 - cos(132.3/2)) = r(1 - 0.404) or about 6/10 the radius of the tank. By changing the fraction a/b, you can mark the dipstick any way you want, although you'll have to go through a fresh round of guessing for each new fraction. But note that you only have to consider fractions up to 1/2, because you can use symmetry to generate the others. Anyway, for a tank with a 10-inch radius (20-inch diameter), 3/4 full is about 6 inches from the top of the tank, and 1/4 full is about 6 inches from the bottom. For variations on this problem, see: Trisecting a Circle with Parallel Cuts - Dr. Math archives http://mathforum.org/library/drmath/view/60807.html Sphere Slices - posted April 29, 2002 Trigonometry and Calculus Problem of the Week - The Math Forum http://mathforum.org/calcpow/solutions/solution.ehtml?puzzle=141 The Rock Garden - posted November 26, 2001 Lucent-Rutgers Problem of the Week - The Math Forum http://mathforum.org/lucentpow/solutions/solution.ehtml?puzzle=57 For solutions sent to Tom and Ray, posted on their site, see: The Great Gas Tank Math Solved http://cartalk.cars.com/Radio/call/dipstick/proof.html The Non-calculus Approach (PDF file) http://cartalk.cars.com/Radio/call/images/click_and_clack.pdf and see Slightly Skewed Solutions http://cartalk.cars.com/Radio/call/dipstick/mail.html - Doctors Ian and Sarah, The Math Forum http://mathforum.org/dr.math/
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