Volume of Tacoma DomeDate: 01/16/2002 at 18:54:19 From: Kacey Beyers Subject: Volume of Tacoma Dome I am trying to figure the volume of the Tacoma Dome in Tacoma, Washington. I know the Diameter is 530'. I know the height to the tallest point of the dome is 152'. The side wall all around the dome is 36' high. I got as far as finding the volume of the base of the dome, which is pi r2 h. = pi (265)(265) h = pi 70225 x 36 = pi x 2528100 The problem I'm having is finding the volume of the lid of the dome. Since it's not a full sphere, I don't really know where to begin. Please give me some advice. Thanks, Kacey Date: 01/17/2002 at 02:42:33 From: Doctor Jeremiah Subject: Re: Volume of Tacoma Dome Hi Kacey, In this answer I am going to use * for multiply to avoid confusion with the letter x, and I will use ^ to mean "an exponent of" so x^2 would be x squared. The shape you're talking about is called a 'spherical cap'. If I understand what you are asking, it is: What is the volume of a spherical cap of height 152 - 36 = 116, whose base has a radius of 530 (where the radius of the sphere is much bigger)? To determine this we need to find R (the sphere's radius). Consider this semi-circle: 530 feet |-----------------| +++++ -+- +++ +++ | 152 - 36 = 116 feet +++---265--+ +++ -+- + \ | / + + \ | / + + \ | / + + R H R + + \ | / + + \ | / + + \ | / + + \|/ + + + + Basically we have a triangle, and Pythagoras' theorem says: R^2 = H^2 + 265^2 and since H = R-116 we have: R^2 = (R - 116)^2 + 265^2 R^2 = (R^2 - 232R + 13456) + 70225 R^2 - R^2 = -232R + 83681 0 = -232R + 83681 232R = 83681 R = 83681/232 feet H = 83681/232 - 116 H = 83681/232 - 26912/232 H = 56769/232 feet This next bit is calculus, and you might not be interested so you could jump right to the end... The volume is calculated by summing up a bunch of really thin disks sitting on top of each other: +++++ --------------------+- +++ +++ | +++-----------------+++ -----+- | + | + | + | dz | +-----+--------/|\--------+-----+ -+- | + | + | R + | + | + + | + + | + + | + R^2 = x^2 + z^2 + --+- + + + + + + + + + + + + + + +++ +++ +++ +++ +++++ z=R / Volume = | Pi*x^2 dz <=== x^2 = R^2 - z^2 / z=H z=R / Volume = | Pi*(R^2 - z^2) dz / z=H z=R z=R / / Volume = | Pi*R^2 dz - | Pi*z^2 dz / / z=H z=H z=R z=R / / Volume = Pi*R^2 | dz - Pi | z^2 dz / / z=H z=H z=R z=R | | Volume = Pi*R^2*z | - Pi*z^3/3 | | | z=H z=H Volume = [ Pi*R^2*R - Pi*R^2*H ] - [ Pi*R^3/3 - Pi*H^3/3 ] But R=83681/232 and H=R-116=83681/232-116=56769/232 Volume = [ Pi*(83681/232)^2*83681/232 - Pi*(83681/232)^2*56769/232 ] - [ Pi*(83681/232)^3/3 - Pi*(56769/232)^3/3 ] Volume = Pi * 12999598/3 Volume = 13613147.194 cubic feet That is the volume of the dome part (not including the base). Now, the total volume is the base volume (Pi * 2528100) plus the spherical cap volume (Pi * 12999598/3) and that is: TotalVolume = Pi * 20583898/3 cubic feet TotalVolume = 21555407.5825 cubic feet That is a lot of volume! You'll find an illustration of a spherical cap in the Dr. Math Geometric Formulas FAQ at: http://mathforum.org/dr.math/faq/formulas/faq.sphere.html That page has the formulas, so you don't have to do it the hard way as I did... Volume = (Pi/6)(3w^2+h^2)h where w=530/2=265 and h=152-36=116 Volume = (Pi/6)(3*265^2+116^2)116 Volume = Pi * 12999598/3 Which is the same answer and is useful if you don't care how to solve it but just want the answer. - Doctor Jeremiah, The Math Forum http://mathforum.org/dr.math/ |
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