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?
