Area and Volume of a FootballDate: 28 Mar 1995 12:08:04 -0500 From: Mary Basse Subject: geometry question Hi. My name is Russell Heinrichs. I am a freshman at Anna-Jonesboro High School. A couple days ago when I was pumping up my football, I thought, "How would one find the area of a football? Or then again, how would one find the volume of a football?" Please help me with these problems if you can. Thanks. Date: 30 Mar 1995 00:42:39 -0500 From: Daniel Eisenbud Subject: Re: geometry question One way to physically find the volume of a football would be to put some water in a container with straight sides (a fish tank, or a big pot, or something) and measure how much the water rises when you submerse the football in it. Then, if you can find the area of the cross-section of the container of water, and multiply that by the increase in height, you'll have the volume of the football. I can't think of any really easy way to physically find the surface area of the football. One way to approach this problem might be to divide the football into one-inch slices from end to end, and measure the how wide the football is from edge to edge (perpendicular to the axis of the football) at some point in each of these slices (for practical purposes, the middle of the slice would be a good place.) Multiply the width of each slice by pi, and you have the circumference of a circle in that slice. Now multiply that by the thickness (in this case 1") of the slice, and you'll get an approximation to the surface area. Now add up the approximations to the surface areas of all the slices, and you'll have something close to the surface area of the football. A theorem in integral calculus says that as the width of the slices approaches zero, and therefore the number of slices approaches infinity, the sum will approach the actual surface area of the football. So you would get a better approximation if you measured every quarter-inch than if you measured every inch. You could do a similar thing to mathematically approximate the volume: find the area of a cross section of each slice, which would be pi*d^2/4 (which is equivalent to pi*r^2), multiply it by the thickness of the slice, for an approximation of the volume of the slice, and add them all up. I hope this answers your question; if you have more questions or anything is unclear (this is hard stuff) please feel free to write back. -Dan "Dr. Math" Eisenbud |
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