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A textbook of Computer Based Numerical and Statiscal Techniques part 4

A textbook of Computer Based Numerical and Statiscal Techniques part 4. By joining statistical analysis with computer-based numerical methods, this book bridges the gap between theory and practice with software-based examples, flow charts, and applications. Designed for engineering students as well as practicing engineers and scientists, the book has numerous examples with in-text solutions. | 16 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Percentage Error in Sr inn 100 r x 100 -------x r r f SR A 2 dR dr Because S _ SR _ 100 SR _ 100 SR h r 2 r 2 fr 2r2 3r h On substituting Percentage Error in r 4.5 and value of SR from 1 100 0.002x50.50 . 0.1x50.50x5.5 --------5-x----------x h -------------- 2 x 4.5 2 11 11x 20.25 0.12 Sh 100 SR 100 SR h x 100 x -----. h h 2 _ h f r 2 1 I 2 W 2 100SX 100 50.5 x 0.002 -----------r -----x------------ 0.505. r 2 1 20 11 11 2F 2 J Example 29. Two sides and included angle of a triangle are 9.6 cm 7.8 cm and 45 respectively. Find the possible error in the area of a triangle if the error in sides is correct to a millimeter and the angle is measured correct to one degree. 1 Sol. Assume that the area of the triangle ABC X bc sin A Error in the measurement of sides and angles are Zb 0.05 cm Zc 0.05 cm and ZA 1 x 0.01745 0.008725 radians 2 ax 1 . c sin A db 2 ax 1 ax 1 b sin A and bc cos A dc 2 dA 2 SX dX Sb s ax Sc sa dX ab dc dc 1111 1 1 0.05 x x 9.6 x- 0.05 x x 7.8 x- 0.008725 x x 9.6 x 7.8 x- 2 y 2 2 V2 2 y 2 1 - 0.05 x 4.8 0.05 x 3.9 0.008725 x 4.8 x 7.8 V2 0.761664 0.5385778 - 0.539 sq. cm. 1.4142135 ERRORS AND FLOATING POINT 17 Example 30. The error in the measurement of area of a circle is not allowed to exceed 0.5 . How accurately the radius should be measured. Sol. Area of the circle nr2 A say dA 2nr dr Percentage Error in A x 100 0.5 A Therefore 0.5 1 2 SA x A nr 100 200 Percentage Error in Sr r - x 100 r 1 2 100 SA 100 200 nr r dA r 2n2 dr 1 0.25. 4 Example 31. The error in the measurement of the area of a circle is not allowed to exceed 0.1 . How accurately should the diameter be measured nd2 Sol. Let d is the diameter of a circle and then its area is given by A . Therefore SA nd dd 2 Since SA Sd therefore Sd Now Percentage Error in SA A x 100 0.1 A Therefore SA 01XA 0.001x A Xnd2 100 4 Similarly Percentage Error in d x 100 d 100 SA 100 0.001 xnd2 2 -----x - x d dA d 4 nd dd 0.1 xndf 2 0.1 ---------x 0.05- 4d