1 CHAPTER 20 20 1 A plot of log10k versus log10f can be developed as 0 7 0 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 0 2 0 0 2 y 0 4224x 0 83 0 4 2 R 0 9532 0 6 As shown the best fit line is log10 f 0 422363log10 k 0 83 Therefore 2 10 0 83 0 147913 and 2 0 422363 and the power model is y 0 147913x 0 422363 The model and the data can be plotted as 1 5 y 0 1479x0 4224 R2 0 9532 1 0 5 0 0 20 40 60 80 100 120 140 160 20 2 We can first try a linear fit y 2 5826x 1365 9 2 R 0 9608 1800 1600 1400 1200 60 30 0 30 60 90 120 As shown the fit line is somewhat lacking Therefore we can use polynomial regression to fit a parabola PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 2 2 y 0 0128x 1 8164x 1331 1800 2 R 0 9934 1600 1400 1200 60 30 0 30 60 90 120 This fit seems adequate in that it captures the general trend of the data Note that a slightly better fit can be attained with a cubic polynomial but the improvement is marginal 20 3 a The linear fit is 80 y 1 05897x 0 81793 R2 0 90583 60 40 20 0 0 20 40 60 80 The tensile strength at t 32 can be computed as y 1 05897 32 0 81793 34 7048913 b A straight line with zero intercept can be fit as y 1 07514x 80 R2 0 90556 60 40 20 0 0 20 40 60 80 For this case the tensile strength at t 32 can be computed as y 1 07514 32 34 40452 20 4 Linear regression with a zero intercept gives note that T K T oC 273 15 PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 3 15000 10000 y 29 728x 5000 R2 0 9999 0 0 100 200 300 400 500 Thus the fit is p 29 728T Using the ideal gas law p V R T n For our fit p 29 728 T For nitrogen n 1 kg 28 g mole Therefore 10 R 29 728 3 8 324 10 28 This is close to the standard value of 8 314 J gmole 20 5 This problem is ideally suited for Newton interpolation First order the points so that they are as close to and as centered about the unknown as possible x0 740 f x0 0 1406 x1 760 f x1 0 15509 x2 720 f x2 0 12184 x3 780 f x3 0 16643 x4 700 f x4 0 0977 The results of applying Newton s polynomial at T 750 are Order 0 1 f x 0 14060 0 14785 Error 0 007245 0 000534 PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 4 2 3 4 0 14838 0 14831 0 14831 7E 05 0 00000 Note that the third order polynomial yields an exact result and so we conclude that the interpolation is 0 14831 20 6 A program can be written to fit a natural cubic spline to this data and also generate the first and second derivatives at each knot Option Explicit Sub Splines Dim i As Integer n As Integer Dim x 100 As Double y 100 As Double xu As Double yu As Double Dim dy As Double d2y As Double Dim resp As Variant Range a5 Select n ActiveCell Row Selection End xlDown Select n ActiveCell Row n Range a5 Select For i 0 To n x i ActiveCell Value ActiveCell Offset 0 1 Select y i ActiveCell Value ActiveCell Offset 1 1 Select Next i Range c5 Select Range c5 d1005 ClearContents For i 0 To n Call Spline x y n x i yu dy d2y ActiveCell Value dy ActiveCell Offset 0 1 Select ActiveCell Value d2y ActiveCell Offset 1 1 Select Next i Do resp MsgBox Do you want to interpolate vbYesNo If resp vbNo Then Exit Do xu InputBox z Call Spline x y n xu yu dy d2y MsgBox For z xu Chr 13 T yu Chr 13 dT dz dy Chr 13 d2T dz2 d2y Loop End Sub Sub Spline x y n xu yu dy d2y Dim e 100 As Double f 100 As Double g 100 As Double r 100 As Double d2x 100 As Double Call Tridiag x y n e f g r Call Decomp e f g n 1 Call Substit e f g r n 1 d2x Call Interpol x y n d2x xu yu dy d2y End Sub Sub Tridiag x y n e f g r Dim i As Integer f 1 2 x 2 x 0 g 1 x 2 x 1 PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 5 r 1 6 x 2 x 1 y 2 y 1 r 1 r 1 6 x 1 x 0 y 0 y 1 For i 2 To n 2 e i x i x i 1 f i 2 x i 1 x i 1 g i x i 1 x i r i 6 x i 1 x i y i 1 y i r i r i 6 x i x i 1 y i 1 y i Next i e n 1 x n 1 x n 2 f n 1 2 x n x n 2 r n 1 6 x n x n 1 y n y n 1 r n 1 r n 1 6 …
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