This problem first came to me in high school, and a couple times since, and I even assigned it for extra credit in one of my calculus classes after I became a teacher. So I know the solution. What I am looking for is other WAYS to obtain the solution. I've been told there exists a solution using only arithmetic, but have never figured it out. Other solutions using ordinary calculus, trigonometry, algebra of conic sections, and so on are also possible.
The problem is usually stated in the form of a letter from an Aunt and Uncle:
Dear niece/nephew, How are things
going for you and your folks? We hear
you are doing quite well it school.
Keep it up! Given this success, we
were hoping you could help us figure
out a little dilemma. As you know,
our home is heated by fuel oil, and we
have a big tank buried in the side
yard. The tank is a cylinder, 20 feet
long and 10 feet in diameter, lying on
its side five feet deep, with a narrow
tube coming to a fill cap at ground
level. Your uncle has a 15 foot
length of old pipe that we'd like to
utilize as a dip stick in order to
know when we are getting close to
needing a fill-up. We know that 0
feet is empty, 5 feet is half full,
and 10 feet is completely full.
Trouble is, we don't know how to mark
any other points. We are pretty sure
they will not be uniformly spaced.
What we really want is to know, within
the nearest 0.01 foot, where to mark
the dip stick for every multiple of
10% from 0% to 100%. Can you figure
this out for us? Of course, we will
want to see details of your solution
and check it ourselves, and it would
especially help if you could draw us a
scale model of the dip stick. Love,
Auntie Flo and Uncle Jim
That last sentence shows the teacher influence on the problem.
So, my challenge to this community is not to find any old solution, but to find the solution at the lowest possible grade level, so to speak.
Thanks.
UPDATE: To those who are focusing in on the .01 feet accuracy, I apologize. The intent was merely to state, it is acceptable to estimate. If the exact answer is sqrt(2)*pi/2 or some other silly thing, go ahead and just write 2.22 feet, for example.
Best Answer
Upper figure: cylinder oil tank cross-section perpendicular to its horizontal axe. The vertical coordinate is the oil level in percentage.
Lower figure: graph of oil volume/max. volume (in %) versus oil level $l$ (in feet). The horizontal straight lines represent the area/volume ratio $A(l)/A(10)=V(l)/V(10)$ (in %) for every multiple of 10% from 0% to 100%.
--
Since the tank radius is $5$, the oil level with respect to the bottom of the tank is given by $l=5-5\cos \frac{\theta }{2}$, where $\theta $ is the central angle as shown in the figure. The area of the tank cross section filled with oil is
$$A(\theta )=\frac{25}{2}\theta -\frac{25}{2}\sin \theta $$
or
$$A(l)=25\arccos (\frac{5-l}{5})-\frac{25}{2}\sin (2\arccos (\frac{5-l}{5}))$$
The area ratio $A(l)/A(10)=V(l)/V(10)$ where $V(l)$ is the oil volume.
Let $f(l)$ denote this area ratio in percentage:
$$f(l)=\frac{100}{\pi }\arccos \left( 1-\frac{1}{5}l\right) -\frac{50}{\pi }\sin \left( 2\arccos \left( 1-\frac{1}{5}l\right) \right) $$
Here is the sequence of $f(l)$ values for $l=0,1,2,\ldots ,10$. The graph of $f(l)$ is shown above.
$f(0)=0$, $f(1)=5.2044$, $f(2)=14.238$, $f(3)=25.232$, $f(4)=37.353$, $f(5)=50$,
$f(6)=62.647$, $f(7)=74.768$, $f(8)=85.762$, $f(9)=94.796$, $f(10)=100$
Edit: One still needs to solve the nonlinear equation $f(l)-10k=0$ for $k=1,2,3,4,6,7,8,9$, e. g. by the Secant Method.
Edit 2: The problem of solving graphically, as shown in the secong figure, is that it would be very difficult, or impossible, to get the required accuracy of 0.01 (feet).
Update: The oil level marks (in feet) should be placed at
$0,1.57,2.54,3.40,4.21,$
$5,5.79,6.60,7.46,8.44,10$
corresponding to the oil volume percentage of
$0,10,20,30,40,$
$50,60,70,80,90,100$.
This calculation was based on the following $f$ function values:
$f(0)=0.0$, $f(1.5648)=10.0$, $f(2.5407)=20.0$, $f(3.40155)=30.0$, $f(4.21135)=40.0$
$f(5)=50.0$, $f(5.7887)=60.0$, $f(6.59845)=70.000$, $f(7.4593)=80.000$,
$f(8.4352)=90.000$, $f(10)=100.0$
Update 2 Figure of marks:
[Rearranged to show the sequence of editions and updates.]