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### 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)

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

The Non-calculus Approach (PDF file)

and see

Slightly Skewed Solutions

- Doctors Ian and Sarah, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus
High School Conic Sections/Circles

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