91Ó°ÊÓ

In a paper dealing with the efficiency of energy utilization of the larvae of the modest sphinx moth (Pachysphinx modesta), L. Schroeder [Schr1] used the following data to determine a relation between \(W\), the live weight of the larvae in grams, and \(R\), the oxygen consumption of the larvae in milliliters/hour. For biological reasons, it is assumed that a relationship in the form of \(R=b W^{a}\) exists between \(W\) and \(R\) a. Find the logarithmic linear least squares polynomial by using $$ \ln R=\ln b+a \ln W $$ b. Compute the error associated with the approximation in part (a): $$ E=\sum_{i=1}^{37}\left(R_{i}-b W_{i}^{a}\right)^{2} $$ c. Modify the logarithmic least squares equation in part (a) by adding the quadratic term \(c\left(\ln W_{i}\right)^{2}\), and determine the logarithmic quadratic least squares polynomial. d. Determine the formula for and compute the error associated with the approximation in part (c). $$ \begin{array}{llllllllll} \hline {c}{W} & {c}{R} & {c}{W} & R & W & R & W & R & W & R \\ \hline 0.017 & 0.154 & 0.025 & 0.23 & 0.020 & 0.181 & 0.020 & 0.180 & 0.025 & 0.234 \\ 0.087 & 0.296 & 0.111 & 0.357 & 0.085 & 0.260 & 0.119 & 0.299 & 0.233 & 0.537 \\\ 0.174 & 0.363 & 0.211 & 0.366 & 0.171 & 0.334 & 0.210 & 0.428 & 0.783 & 1.47 \\\ 1.11 & 0.531 & 0.999 & 0.771 & 1.29 & 0.87 & 1.32 & 1.15 & 1.35 & 2.48 \\ 1.74 & 2.23 & 3.02 & 2.01 & 3.04 & 3.59 & 3.34 & 2.83 & 1.69 & 1.44 \\ 4.09 & 3.58 & 4.28 & 3.28 & 4.29 & 3.40 & 5.48 & 4.15 & 2.75 & 1.84 \\ 5.45 & 3.52 & 4.58 & 2.96 & 5.30 & 3.88 & & & 4.83 & 4.66 \\ 5.96 & 2.40 & 4.68 & 5.10 & & & & & 5.53 & 6.94 \\ \hline \end{array} $$