Help trucker Rich measure his gas

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121

The answer is 0.596027 x the distance of the half tank left distance.

Essentially this can be reduced to the equation for an Area of a Chord since
the volume of a cylinder here can be viewed as a cylinder of length 0.

It is worked out nicely at the Wolfram Math World site below where the area of a 1/4 of a circle of radius R (R being the half full tank measurement) is used to solve the equation.

http://mathworld.wolfram.com/CircularSegment.html

Quoting from the page : NOTE h is the 1/4 full distance and R is the half full distance.

" To find the value of h such that the circular segment has area equal to 1/4 that of the circle, plug A=piR^2/4 into equation (18) and divide both sides by R^2 to get
1/4pi=cos^(-1)(1-x)-(1-x)sqrt(2x-x^2),
(23)

where x=h/R. This cannot be solved analytically, but the solution can be found numerically to be approximately x=0.596027… (Sloane’s A133742), corresponding to h=0.596027R. "

Pete Costello

122

@so_far_so_good congrats on a clever C program! But I don’t understand your deprecation of engineering. Engineering is simply the use of math to solve a real world problem. Exactly what is called for here. There are many ways to skin the mathematical cat including geometry, trig, calculus, number theory etc. Use whatever you’re comfortable with that gets you to a usable answer. All the methods proposed in the forum are creative and good (except for the creative and wrong ones). For some reason, many people find calculus and numerical integration scary and hard, but once you get the hang of it, it’s quite handy and useful. In fact, you have used numerical integration in your program. As Ray said, it’s just area under the curve. Maybe using a package like VisSim is a bit of overkill for this particular application, but numerical integration is very useful tool that is essential for modern technology like aircraft, orbiting satellites, power plants, engine control units, water treatment, econometric models, weather prediction etc. Numerical integration tools like VisSim should be part of everyones problem solving tool kit.

123

I
think I remember that the tank was 20"
diameter. This gives an area of 314 square
inches. A half tank will be 10 inches down
and have an area of 157 square inches. Now,
without Calculus, you can estimate. divide
the 157 by 2 to get the quarter tank. Giving
77.5 square inches. Now, divide that by
about 20 inches to come up with about 4
inches below the half way mark will give you
a little less than a quarter tank. Sooo, 6
inches from the bottom should do it!

124

This is the answer:
A cylindrical tank 1/4 full is when the stick length is 0.298 of that for the full tank.

To get this result, first note that this area is given in radians by RR/2(angle-sin(angle)). If we set this equal to PIRR/4, we get angle(radians)=2.31031 which is ~ 132 degs. From the bottom, the corresponding height is R*(1-cos(angle/2)). If we divide this by the diameter (2*R), we get 0.298.

125

20 inch fuel tank

I know there are more complicated solutions, but the end
result involves a stick with a notch in it.

I fired up Google sketch-up, made a 20 inch diameter circle.
Asked for the total area, and then started interpolating
various levels and recorded the resulting areas.

At 6’ from the bottom I got very close to 25% of the total
area.

126

Last week the trucker that called with the
fuel tank problem ( 1/4 calibration stick…)
does not have to search for complex math to
solve his problem. My common sense method
is simple, quick and accurate. Here goes:

  1. fill the tank with fuel. Mark the
    mileage. Keep checking the tank dip stick
    until you have burned half the tank.
  2. record the mileage. Subtract the mileage
    from the start value, then cut this figure
    in half. THIS IS THE 1/4 tank mileage.
  3. Drive exactly this number of miles more,
    pull over and mark the dipstick at 1/4 !.
    THAT IS IT ! If you need to mark, the 3/4
    point, fill the tank, drive the mileage and
    pull the stick and mark it 3/4. Voila !
127

This was a really interesting call-in question because it’s an issue of getting him a quick answer rather than a precise one. Yes, a precise answer requires integral calculus, but you don’t need to give a guy with a wooden dipstick a precise answer. We can get a decent round answer using some simple geometry and extrapolation.

He actually provided a good hint by saying “I know I don’t have a quarter tank left when the dipstick measures 15”. Well, how much of a tank does he have left? Well, let’s use simple geometry to make a quarter-circle cross-section. At 15", you’re at 1/3 of the way down the arc of the cylinder (30 degree right triangle geometry), so take a third of that quarter-circle, and add to that the area of the triangle underneath, and divide it by the total area of the quarter-circle.

It took me about a minute of chicken-scratch to get 0.61. Translate that to the full tank, and you have a little less than a fifth of the tank left at 15" of the dipstick. We can then do a rough extrapolation to say, well, at 14" you have about a quarter of the tank left.

He can take the rest of the equation to his local university, or save everybody’s time by taking his car to a mechanic to get the gas gauge serviced. I know a guy who fixed this on his '65 mustang in about five hours.

128

Cylinder Tank Dip Stick Gauge

I assume the 18 wheeler guy had two identical
tanks on his truck. I also assume he knows how
many gallons a tank holds when it is full.
Why not empty one tank, then fill it with 25%
of the total and then drop the stick in and
measure the height. He could then add 50% more
and find the 3/4 amount. I liked math in
school but this seems like a good way to do it.
Just thinkin. Thanks for the show. My brotha
lives in Medford and each week I think of him
when I listen to you. And, no, I don’t drive
like my brotha.

129

Attached is a paper that solves this problem exactly. The diameter of the tank is 20". The 1/4 mark corresponds to 6" measured on the dowel. The 3/4 mark corresponds to 14" measured on the dowel.

Cheers,

Mike

130

The answer to the caller’s question can be found by finding
the centroid of a semi-circle with a 10 inch
radius. The formula is y = 4r/3pi where y is
the distance from the center of the complete
circle. So the centroid of each half circle
is 4.24 inches from the center of the
cylindrical tank. To measure 1/4 full, the
measurement from the bottom of the tank is
10-4.24 or 5.76 inches.

He could also find the answer without using
math by cutting a 10 inch radius semi-circle
out of corrugated cardboard or thin plywood.
The centroid is the point where the semi-
circle will balance on the end of something
like a round dowel.
Or, fill the tank, syphon out 1/4 of the easily calculated volume, take a dipstick measurement. The distance to the 1/4 full mark is 20 inches minus this measurement.

131

Your trucker who phoned with the inoperable fuel gage and his drip stick problem needs to know less how many gallons of diesel he has left, and more importantly, how many miles he can go on the remaining fuel. That’s easy. Fill up and drive a few hundred miles. Stop, fill up again and calculate MPG, you don’t need calculus for that, just division.

He can then dive 100 miles, stop, dip his stick and mark off 100 miles. He can do this every few hundred miles if he wants, but his odometer on the dash should give him a good indication of miles remaining in the tank, once he knows MPG.

Knowing where 3/4, 1/2 or 1/4 of a tank is less important, though an interesting problem to calculate.

I have a boat with 150 gallon tank. I purposely ran out of fuel at the dock a few years ago just so I could fill up, stopping every 25 gallons to mark my dip stick. Now I know how much fuel I have remaining and how far I can expect to motor on what’s left because I know my MPG. It?s a sail boat, but this was important last yea. We had to motor for 4 days between Bermuda and the Virgin Islands. It’s a total of 860 miles across the open ocean, no filling stations out there. We had to motor fore more than half the distance, but we had enough fuel of 700 miles. That’s what we needed to reach the trade winds and begin sailing again before running out. We were in touch with another yacht, adrift half way to the Island that had un out of fuel. By knowing my MPG and using my dip stick I calculated I had enough fuel in the tank to get us to the Trades, and could spare the 30 gallons I had on decks in jerry cans. We met the crew on the other boat at mid-night in the middle of the Bermuda Triangle and transferred our spare jerry cans, one at a time, to the disabled yacht. They made it to the Islands and so did we.

Knowing your MPG can be helpful. My Chevy Tahoe does his for me automatically.

David Lyman, Capt. Searcher, a Bowman 57 ketch
Rockport, Maine

132

To satisfy the caller, Rich, I submit this table. It features half-inch increments:

Height(in) % Full
0.00, 0.0
0.50, 0.6
1.00, 1.8
1.50, 3.4
2.00, 5.2
2.50, 7.2
3.00, 9.4
3.50, 11.7
4.00, 14.2
4.50, 16.8
5.00, 19.5
5.50, 22.3
6.00, 25.2
6.50, 28.1
7.00, 31.1
7.50, 34.2
8.00, 37.3
8.50, 40.4
9.00, 43.6
9.50, 46.8
10.00, 50.0
10.50, 53.1
11.00, 56.3
11.50, 59.5
12.00, 62.6
12.50, 65.7
13.00, 68.8
13.50, 71.8
14.00, 74.7
14.50, 77.6
15.00, 80.4
15.50, 83.1
16.00, 85.7
16.50, 88.2
17.00, 90.5
17.50, 92.7
18.00, 94.7
18.50, 96.5
19.00, 98.1
19.50, 99.3
20.00, 100.0

133

Connie B. suggested I add my 2?s. Below is my email I sent on Saturday after the show.
Level of gas on dowel

Dear Click & Clack,

I was intrigued by the question asked by the
trucker as to what level on a dowel would
the gas level need to reach to measure a ?
full on a cylindrical (saddle) gas tank . I
told my wife that was straight forward and
I?d solve it when I get home. As you guys
said it was just integral calculus. So when
I got home I was set on spending 15 minutes
to solve it. All I had to do was apply
integral calculus to the equation of a
circle. But I forgot how to integrate a
square root function ? my prowess as an
engineer was at stake. Not to worry -I
Googled the equation. It was there ?
actually several solutions. Wow! I used
Excel to interpolate. Not elegant but ? The
gas should reach 30% of the height of the
tank (diameter) to ensure a tank 25% full.

Sincerely,
Jeff Feld
An Avid Car Talk Listener

To be more precise and assuming the bottom of the dowel is honed to a point, the answer I found was 29.8% * diameter of tank, i.e., the gas should reach 5.96? from the bottom of the dowel to be ? full.

134

I may be looking at this too simply, but the dash guage never told Rich what volume of fuel he had left in the tank, it told him the fuel level. The way the fuel tank sender works is it is a float device that as it floats at the very top the tank the dash guage shows full, when the tank is half empty it shows 1/2 tank which is 1/2 the height of the fuel in the tank ( which at 1/2 tank is also 1/2 of the volume of course) and half the distance between 1/2 tank and empty would be 1/4 tank, not 1/4 of the volume of the tank but 1/4 the height of the fuel in the tank. The guage measures level NOT volume.

135

This is a geometry/trigonometry problem. So it doesn’t require the high power of intregral calculus.

I took the simple way. I looked in the CRC Handbook of Chemistry & Physics, 49th edition and found the formula for the area of a segment of a circle on page A-257. With a programable calculator and the entry of a few trial numbers, I had the distance from the bottom to mark the dowel. As seems to be the consensus and I concur, the answer is 6 inches (5.96 if you want to be rigorous) for 1/4 of the tank volume and 14 inches (14.04 inches) for 3/4 tank full.

I am enheartened to see all the replies that have been presented. Keep up the good work Tom and Ray (aka Tappet Brothers).

136

Semi trucks don’t have fuel selector switches in the cab, the tanks freely flow back and forth to equalize. I suppose if they didn’t, when you pulled in for fuel, you’d have to pull up next to the pump on one side, then turn your 60+ foot rig all the way around and pull up on the other side. Much easier to pump fuel into one tank only.

137

A number of people have already posted the correct answer (5.96" for 1/4, and 14.04" for 3/4), but here’s something to stick on the ol’ dashboard…

138

Finding 1/4 tank marking point on stick

I can’t believe you guys! You want to know
where to mark the 1/4 tank point on the
truckers stick (your 11/6/2010 show)? The
trucker knows the size of his tank – the
total number of gallons it holds. All he has
to do is use the gas in his tank down to
almost nothing left (I suggest when he knows
he is getting low he drives around the
perimeter of a truck stop). When he gets the
tank low to the point that the stick shows
just a bit of gas on the end of the stick, he
drive to the nearby pump and fills his tank
with .25 x total gallons his tank holds. Then
he puts the stick in the tank, pulls it out
and marks the point where the quarter full
tank comes to. Right?!

139

The bottom line is: If zero inches on the dipstick is empty and 20 inches is full,
the quarter tank mark is at 5.96 inches (10 - 4.04) and the three-quarter mark is at 14.04 inches (10 + 4.04).

I can think of at least 5 ways to approach this problem (none involving asking around at a truck stop “Who has a quarter tank?”).

  1. If possible, let one saddle tank drain completely empty. If the trucker knows the capacity of the tank, he can just put one quarter that amount into the tank and see what the dipstick shows.

  2. Get a straight-sided (not tapered) cylindrical jar of known capacity, e.g. a 16 ounce jar. Put 4 ounces of water into the jar, close the top and put it on its side. Then measure the height of the water line. From here it’s a simple ratio problem.

  3. Scale drawing approach: Using a compass and a sheet of graph paper, draw a circle having a radius of 10 grid units. Make sure that the center of the circle is at the intersection of two grid lines. The area of this circle is 100 pi, or 314.1692 squares. Darken the horizontal diameter line (it should be along one the grid lines), dividing the circle into two equal halves. Start counting and marking the squares adjacent to and above this diameter line (the first row should have 20 squares). Count only squares that are at least half inside the circle. When you’ve counted 78 squares (roughly 314 divided by 4) you should have about four rows of squares shaded. Four squares is (to scale) 4 inches above the half-full line.

But for the more mathematically inclined, the solution could involve integral calculus as was mentioned on your program.
If we have a unit circle centered at the origin, the area under the semicircular curve y = sqrt ( 1 - x-squared) should give us an area of pi over 2. Then we can try moving the center of the circle down by some displacement “d” and try to get a curve over an area of pi over 4. I realized that I was too rusty at calculus to solve this whole thing analytically, so I resorted to the skills I am more practiced at: computer programming. I then tried the following two approaches:

  1. Numerical integration approach 1. This is the computer equivalent to the scale drawing approach (3 above) but with as fine a grid as we care to use. We just start counting squares on a fine grid, row by row, until we’ve got an area of pi / 4. Or to make greater use of symmetry, we just count squares in the first quadrant (x and y non-negative) until we have and area of pi / 8.

I wrote a simple ‘C’ program to do this calculation… see attached file.

  1. Numerical integration approach 2, with assistance from Wolfram software.

http://integrals.wolfram.com/index.jsp

This is probably more complicated than the problem warrants. But if you input the formula (sqrt [ 1.0 - xx] - d) into the Wolfram integrator, it will give you an antiderivative of 0.5 * ( -2dx + sqrt (1 - xx) + arcsin(x) ) + constant. If we fiddle with various values for ‘d’ we can eventually find that the area under the curve (sqrt [ 1.0 - x*x] - d) is pi/4 when d is about 0.404.

Best regards,
Tom in Gilberts, IL

140

1/4 gas tank stick

In response to the question regarding the gas tank and a stick to measure 1/4 of the tank.
First, you were correct in that you need only calculate for the area of a circle, volume extension is irrelevant.
Solution: The area (A) of a circle or radius “r” is pir^2; the area of the half circle is
(pi^r^2)/2. Similarly, the area (B) of a half circle of radius “x” is (pi*x^2)/2. Set
A = 2B and solve for x. x=r/sqrt2.