Click here to load reader

• date post

16-Oct-2014
• Category

## Documents

• view

69
• download

0

Embed Size (px)

### Transcript of Mathematica 3D Model

Volumes of RevolutionIn this project, I found the volume of revolution of a margarita glass using a digital caliper in order to find themeasurements. I took measurements every 3mm apart in height, and measured the diameter of the glass at eachheight. This is the data, shown {height, width2}:f Table]]]0,80.442), ]3,78.942), ]6,60.192), ]9,42.882), ]12,25.032), ]15,10.652),]18,10.402), ]21,10.422), ]24,10.582), ]27,10.722), ]30,10.902), ]33,11.002),]36,11.022), ]39,11.222), ]42,11.402), ]45,11.472), ]48,11.672), ]51,11.992),]54,12.072), ]57,12.112), ]60,12.242), ]63,12.622), ]66,12.682), ]69,12.932),]72,12.972), ]75,14.352), ]78,17.202), ]81,21.412), ]84,25.582), ]87,29.512),]90,33.732), ]93,37.252), ]96,40.312), ]99,43.662), ]102,46.072),]105,48.172), ]108,49.962), ]111,51.412), ]114,52.622), ]117,53.972),]120,55.612), ]123,56.782), ]126,67.602), ]129,77.092), ]132,84.532),]135,90.812), ]138,97.862), ]141,103.652), ]144,108.702), ]147,112.392),]150,114.572), ]153,116.802), ]156,118.802), ]159,120.142), ]162,121.292));Trapezoidal RuleThe area for the trapezoid is: b an f x0 f x1 f x2 ... f xn1 f xn.In this case, b=162, a=0, and n=54. But since we are trying to find the volume of the glass not just the area, wemust use the disk method, multiplying everything by and squaring each measurement. So,In[4]:=162 54 ](80.44 / 2)2 (78.94 / 2)2 (60.19 / 2)2 (42.88 / 2)2 (25.03 / 2)2

(10.65 / 2)2 (10.40 / 2)2 (10.42 / 2)2 (10.58 / 2)2 (10.72 / 2)2

(10.90 / 2)2 (11.00 / 2)2 (11.02 / 2)2 (11.22 / 2)2 (11.40 / 2)2

(11.47 / 2)2 (11.67 / 2)2 (11.99 / 2)2 (12.07 / 2)2 (12.11 / 2)2 (12.24 / 2)2

(12.62 / 2)2 (12.68 / 2)2 (12.93 / 2)2 (12.97 / 2)2 (14.35 / 2)2 (17.20 / 2)2

(21.41 / 2)2 (25.58 / 2)2 (29.51 / 2)2 (33.73 / 2)2 (37.25 / 2)2 (40.31 / 2)2

(43.66 / 2)2 (46.07 / 2)2 (48.17 / 2)2 (49.96 / 2)2 (51.41 / 2)2 (52.62 / 2)2

(53.97 / 2)2 (55.61 / 2)2 (56.78 / 2)2 (67.60 / 2)2 (77.09 / 2)2 (84.53 / 2)2

(90.81 / 2)2 (97.86 / 2)2 (103.65 / 2)2 (108.70 / 2)2 (112.39 / 2)2

(114.57 / 2)2 (116.80 / 2)2 (118.80 / 2)2 (120.14 / 2)2 (121.29 / 2)2) // NOut[4]= 453080. The Trapezoidal Rule with the disk method gives us an approximated volume of 453, 080 mm3, or 453.08 cm3.Simpsons RuleThe area for Simpsons Rule is: ba3 n f x0 4 f x1 2 f x2 4 f x3 ... 4 f xn1 f xn.In this case, b=162, a=0, and n=54. But since we are trying to find the volume not just the area, we must use thedisk method, multiplying everything by and squaring each measurement. So,Simpsons RuleThe area for Simpsons Rule is: ba3 n f x0 4 f x1 2 f x2 4 f x3 ... 4 f xn1 f xn.In this case, b=162, a=0, and n=54. But since we are trying to find the volume not just the area, we must use thedisk method, multiplying everything by and squaring each measurement. So,In[2]:=162 162

](80.44 / 2)2 4 (78.94 / 2)2 2 (60.19 / 2)2 4 (42.88 / 2)2 2 (25.03 / 2)2 4 (10.65 / 2)2

2 (10.40 / 2)2 4 (10.42 / 2)2 2 (10.58 / 2)2 4 (10.72 / 2)2 2 (10.90 / 2)2

4 (11.00 / 2)2 2 (11.02 / 2)2 4 (11.22 / 2)2 2 (11.40 / 2)2 4 (11.47 / 2)2

2 (11.67 / 2)2 4 (11.99 / 2)2 2 (12.07 / 2)2 4 (12.11 / 2)2 2 (12.24 / 2)2

4 (12.62 / 2)2 2 (12.68 / 2)2 4 (12.93 / 2)2 2 (12.97 / 2)2 4 (14.35 / 2)2

2 (17.20 / 2)2 4 (21.41 / 2)2 2 (25.58 / 2)2 4 (29.51 / 2)2 2 (33.73 / 2)2

4 (37.25 / 2)2 2 (40.31 / 2)2 4 (43.66 / 2)2 2 (46.07 / 2)2 4 (48.17 / 2)2

2 (49.96 / 2)2 4 (51.41 / 2)2 2 (52.62 / 2)2 4 (53.97 / 2)2 2 (55.61 / 2)2

4 (56.78 / 2)2 2 (67.60 / 2)2 4 (77.09 / 2)2 2 (84.53 / 2)2 4 (90.81 / 2)2

2 (97.86 / 2)2 4 (103.65 / 2)2 2 (108.70 / 2)2 4 (112.39 / 2)2 2 (114.57 / 2)2

4 (116.80 / 2)2 2 (118.80 / 2)2 4 (120.14 / 2)2 (121.29 / 2)2) // NOut[2]= 428315.Simpsons Rule with the disk method gives us an approximated volume of 428, 315 mm3, or 428.32 cm3.This graph shows the data points, representing the glass turned sideways and split in half. We need to split theglass in half symetrically so that it can be rotated around the x-axis in order to create the 3D object, shown furtherdown, and to find the volume of revolution.ListLinePlot[f]50 100 150102030405060And here is a 3D representation of the object:g Interpolation[f, InterpolationOrder 1]InterpolatingFunction0., 162., 2 Untitled-3.nbParametricPlot3D[{x, g[x] Cos[t], g[x] Sin[t]}, {x, 0, 162},{t, 0, 2 }, AxesOrigin {0, 0}, Boxed False, AxesLabel {x, y, z}]050100150x50050y50050zUntitled-3.nb 3