1. In studies of radiation-induced polymerization, a source of gamma rays was em
ID: 1721316 • Letter: 1
Question
1. In studies of radiation-induced polymerization, a source of gamma rays was employed to give measured doses of radiation. However, the dosage varied with position in the apparatus, with these figures being recorded. 40 Position, T0 1.0 2.0 in. from base point 1.90 2.74 3.20 2.98 3.20 Dosage, 10s radshr For some reason, the reading at 2.5 inches not reported, but the value of radiation there is needed. Fit interpolating was polynomials of various degrees to the data too supply the missing information. What do you think is the best estimate for the dosage level at 2.5 inches?Explanation / Answer
1) Using the 6 points
(1,2.71) (1.5,2.98) (2,3.20) (3,3.20) (3.5,2.98) (4.0,2.74),
Lagrange interpolation polynomial is
f(x)=0.004x50.00533333x40.201667x3+0.738333x20.357333x+2.532
Using this f(2.5) = 2.504
2) Using the 4 points (1.5,2.98) (2,3.20) (3,3.20) (3.5,2.98)
Lagrange interpolation polynomial is
f(x)=0.x30.293333x2+1.46667x+1.44
f(2.5) =3.27363
Note that even though 4 points were input, it returned a quadratic (not a cubic as would be normal)--capturing the symmetry of values about the point 2.5
It is reasonable to estimate the value at 2.5 to be 3.2 in view of the symmetry of values from 1 to 4 about the point 2.5
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