Volume of Liquid in a CylinderDate: 6/4/96 at 3:17:54 From: Bill Billman Subject: Special considerations to cylindrical measurements Dr. Math, I am experiencing a problem in understanding how to calculate the area under a chord of a circle. Actually, the problem involves the amount of liquid in a cylinder that is laid in the horizontal plane. With a cylinder lying horizontally, I would like to know the partial liquid contents if I can determine the height of the liquid. Example: Let's say there is a cylinder 100 inches in diameter and 200 inches in length. Knowing the diameter of the circle (100 inches) and the length (200 inches), is there a formula that will yield the amount of liquid when the tank has 2, 3, 4 .... standing inches in it? If possible, please advise, and thank you for your time.... Bill Billman Date: 6/4/96 at 8:14:14 From: Doctor Anthony Subject: Re: Special considerations to cylindrical measurements The area of cross-section of the liquid is found by calculating the area of the sector of the circle in which the liquid lies, and subtracting from this the area of the triangle formed by the surface of the liquid and the radii from the centre of the circle to the edges of the surface. If O is the centre of the circle and AB the chord, then the required area of cross-section of liquid is given by: area of sector OAB - area of triangle OAB Taking your figures with radius of circle 50 inches, and shall we say a depth of 10 inches of liquid, we first find the angle AOB. This is done by finding half its value (say X) and then calculating the areas of the sector OAB and the triangle OAB. cos(X) = (50-10)/50 = 40/50 and so X = 36.87 degrees 2X = 73.74 degrees = 1.287 radians Formula for area of sector = (1/2)*(radius^2)*(angle in radians) Area of sector = (1/2)*(50^2)*1.287 = 1608.75 Area of triangle = (1/2)*(OA*OB)*(sin(AOB)) = (1/2)*(50^2)*0.960001 = 1200.00125 Subtracting these two areas we get 408.75 sq inches The volume of liquid is then found by multiplying this area by the length of the cylinder (= 200 inches) Required volume = 408.75*200 = 81749.75 cubic inches -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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