Water in a Horizontal Tank

Date: 8/7/96 at 18:55:50
From: Michael Sheena
Subject: Volume of Water in Side-ways Cylinder

Hi,

I'm trying to find the partial volume (in gallons) of a cylinder-like 
tank that is 72" long AND 36" diameter.  Here's what I used to find 
the TOTAL volume (I am assuming that 231 cubic inches equal a gallon):

((pi*324)72)/231 = 317.26 gallons

But now let's say that the cylinder is only filled up to 4.25"  In 
other words, the diameter of the tank being 36", only 4.25" of that is 
filled with liquid running the entire 72" length of the tank.  Do I 
halve 4.25" and plug that radius into that equation, which would equal 
4.42169 gallons?  Do I shorten the tank from 72" to 4.25", which would 
equal 17.7272 gallons?  Or do I divide 4.25" by 36" to get a 
percentage of the tank and then take that percentage of 317.26 gallons 
for a total of 37.45430556 gallons? 

Or am I all wet?

Date: 8/7/96 at 20:45:22
From: Doctor Robert
Subject: Re: Partial Volume of a Cylinder

In other words, you are trying to find the liquid volume of a cylinder
whose axis is parallel to the ground.  I worked out this problem many
years ago and it involves a lot of calculus.  Here is the general way
to work it.  

Look at the end face (the circular face) of the cylinder and try to 
calculate the area which is covered by the liquid.  Then multiply this 
area by the length of the cylinder.  I believe that you'll need some 
calculus to find that area.  

The volume is definitely not a linear function of the height, because 
the cross section is circular, not rectangular, so taking a percentage 
of the diameter would not give the correct answer.  If you want a 
calculus answer, please write back.

-Doctor Robert,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/