A Cylindrical Tank Problem
A tank is shaped like a cylinder lying on its side. It has a radius of 3 and a length of
10. Here is a picture of it end-on, showing a horizontal section h units above the
center, of thickness dh:
h
R=3
Suppose we take the origin to be the center of the circle, so that the bottom of the
tank has h  R and the top of the tank has h  R. Suppose we fill the tank to a total
depth of L. Then this corresponds to h  L  R.
Problem 1: Find the volume of the tank as a function of L. (The best way to do this is
to find this volume in terms of h then substitute h  L  R  L  3.) You have to be very
careful in setting up this integral. Note, following the diagram above, that the thin
rectangular cross-section shown has width 2 9  h 2 . It creates a thin "slab" of volume:
dV 2
9  h2
Width
 10  dh .
 
Length
Thickness
Your integral will "add up" all these slabs. Check your formula by seeing if the volume
is 0 when L  0, and equals the volume of the cylinder when L  6. (these correspond
to h  3 and h  3.)
Problem 2: Plot volume V as a function of depth L.
Problem 3: Find, correct to 5 decimal places, the level L which makes the volume 240
(cubic units).
Problem 5: Fuel tanks such as ones used for home heating oil have "dipsticks" that
are inserted through a hole in the top, and that measure the amount of oil in the tank
(by measuring L). Explain how you might calibrate such a stick. Since these do not
have to be very accurate in practice, what does the shape of the graph from problem 2
suggest about a quick calibration?