Volume of Spherical CapDate: 01/29/2002 at 14:09:45 From: Roger Staiger Subject: Cone and sphere There is a problem I cannot do; perhaps you can assist. If a heavy sphere, whose diameter is 4 inches, be put into a conical glass full of water whose diameter is 5 and altitude 6 inches, how much water will run over? Ans: nearly 35/47 of a pint. There were two kinds of pints in use back then: the "wine pint," which equalled 1/8 of a "wine gallon" or 28.875 (231/8) cubic inches, and the "ale pint," which equalled 1/8 of a "ale gallon" or 35.25 (282/8) cubic inches. Perhaps an answer in cubic inches is more useful since it avoids the units problem; but I did want to give you the answer. Your assistance is appreciated. Roger Staiger Date: 01/29/2002 at 15:11:19 From: Doctor Peterson Subject: Re: Cone and sphere Hello again! I think the text must have taught the formulas for a spherical cap; this one from the Dr. Math Geometric Formulas FAQ uses the volume formula http://mathforum.org/dr.math/faq/formulas/faq.sphere.html#spherecap V = (Pi/6)(3r_1^2+h^2)h where r_1 is the radius of the cross-section, and h is the distance from the center. Now consider the picture: | | |<-------R------>| | | | ooooooooooo | ooo | ooo | oo D| r_1 oo F E ---- +-----+----------+----------+-----+ ^ \ o |h / o / | \ o | / o / | \o C+ r o/ | o | \ o | o | + | o | o B | oo | oo H \ooo |d ooo/ | \ ooooooooooo / | \ | / | \ | / | \ | / | \ | / | \ | / | \ | / v \|/ --------------------- + A Here R = 5/2, H = 6, r = 2. To find d, the height of the center of the sphere from the bottom of the cone, note similar triangles ABC and ADE: d/r = AE/R = sqrt(R^2 + H^2)/R Therefore d = r/R sqrt(R^2 + H^2), and h for our volume formula is H-d: h = H - r/R sqrt(R^2 + H^2) To find r_1, use the Pythagorean theorem on triangle CDF, giving r_1^2 = r^2 - h^2 I'll leave the rest to you: find the volume of the spherical cap BELOW the level of E, and that is the amount the spills. I got 0.7453, or about 35/47, of an ale pint. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 01/29/2002 at 17:26:52 From: Roger Staiger Subject: Geometry problem Thank you again for solving my grandfather's grandfather's father's (his name was John E. Spare of Pottstown, PA) problem. I think there is another explaination for the spherical cap issue. I think it is less likely the text had an emphasis on spherical caps; and more likely that I never studied or used spherical cap. I don't ever remember studying spherical caps in my life. I don't ever think there was a need for spherical caps in my electrical engineering career. Thanks again. Your help was appreciated. Please write if I can ever reciprocate. Roger Staiger Date: 01/29/2002 at 22:38:17 From: Doctor Peterson Subject: Re: Geometry problem Hi, Roger. Did I say they emphasized spherical caps? I agree with you, just mentioning them at all makes his old text different from our experience. I don't think I learned about them before joining Dr. Math! One reason I'm interested in your problems is that my family has a similar collection of schoolbooks and diaries from my great grandfather (and his father) going back only to the 1870's, but still of great historical and personal interest. To see how much more, in some ways, and less in others, the ordinary high school student learned back then gives an interesting perspective, doesn't it? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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