Math CentralQuandaries & Queries


Question from Mario:

I want to determine how many gallons i have inside a cylinder (tank) that is resting on its side (the Height), NOT standing up.
I know V=pi x r2 x H. And 1 cubic foot = 7.48 gallons. Here are the dimensions r=2' H=20'. Now my question is how do I determine how much
liquid i have inside, if the level of the liquid is about 1/2 of the way of its Diameter (in other words 2'). Remember this cylinder is lying
on its side.
Thank you very much.

Hi Mario,

If the level of the liquid is half way up, that is 2 feet then the tank is half full so the amount of liquid in the tank is

1/2 π r2 h = 1/2 π 22 × 20 = 125.66 cubic feet

which is

125.66 × 7.48 = 940 gallons.

If you want an expression for the volume of liquid in the tank as a function of the depth d at the centre you can extract that from the response that Stephen gave to a similar question by Andy. Andy's tank had a radius of 5 feet and a length of 35 feet. If you have a tank of radius r feet and height h feet then tracing through Stephens development you find that the volume at a depth of d feet (d less than r) is

volume cubic feet.

Make sure that when you use cos-1 on your calculator that it is set in radian mode and not degree mode.

I hope this helps,


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