Numerical Methods For Engineers, 7 Ed
Numerical Methods For Engineers, 7 Ed
7th Edition
ISBN: 9789352602131
Author: Canale Chapra
Publisher: MCGRAW-HILL HIGHER EDUCATION
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Chapter 20, Problem 1P

Perform the same computation as in Sec. 20.1, but use linear regression and transformations to fit the data with a power equation. Assess the result.

Expert Solution & Answer
Check Mark
To determine

To calculate: A power equation fit to the data for specific growth rate and available food the data using linear regression equation and transformation.

Answer to Problem 1P

Solution: The power equation to fit the data for specific growth rate and available food is k=0.1479f0.4224.

Explanation of Solution

Given Information: Data for specific growth rate k and available food f is given in the form of a table as,

fk70.2990.37150.48250.65400.80750.971000.991501.07

Formula used: A linear regression equation is given as y=ax+b.

Here,

a=ni=1nxiyij=1nxjk=1nykni=1nxi2(i=1nxi)2 and b=1n(i=1nyiaj=1nxj).

Power equation can be obtained from the linear regression equation as,

y=αxβ

Here, α=10b and β=a.

Calculation: For the given data, n=8 as there are eight observations.

Consider logarithmic terms and evaluate the summationsas,

logflogklogflogk(logf)20.850.540.450.710.950.430.410.911.180.320.371.381.400.190.261.951.600.100.162.571.880.010.023.522.000.000.014.002.180.030.064.7412.031.561.6319.78

Calculate the value of a as,

a=ni=1nxiyii=1nxii=1nyini=1nxi2(i=1nxi)2

Here, x is logf and y is logk.

Thus,

a=ni=1nlogfilogkii=1nlogfii=1nlogkini=1nlogfi2(i=1nlogfi)2

Simplify,

a=nlogflogklogflogknlogf2(logf)2

Substitute the values from the summation table and solve as,

a=8×(1.63)(12.03)(1.56)8×(19.78)(12.03)2=0.4224

Calculate the value of b as,

b=1n(i=1nyiaj=1nxj)

Here, x is logf and y is logk.

Thus,

b=1n(i=1nlogkiaj=1nlogfj)

Simplify,

b=1n(logkalogf)

Substitute the values from the summation table and solve as,

b=18(1.56(0.4224)(12.03))=0.8300

Thus, the linear regression equation is,

logf=0.4224logk0.83

Therefore, α=100.83 and β=0.4224.

Thus, the power equation fit to the datais written as,

k=0.1479f0.4224

The plot for both the linear and the power models is obtained as,

Hence, the power equation fit gives better approximation than the linear regression fit.

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Chapter 20 Solutions

Numerical Methods For Engineers, 7 Ed

Ch. 20 - Prob. 11PCh. 20 - The molecular weight of a polymer can be...Ch. 20 - 20.13 On average, the surface area A of human...Ch. 20 - 20.14 Determine an equation to predict metabolism...Ch. 20 - 20.15 Human blood behaves as a Newtonian fluid...Ch. 20 - 20.16 Soft tissue follows an exponential...Ch. 20 - 20.17 The thickness of the retina changes during...Ch. 20 - 20.18 The data tabulated below were generated from...Ch. 20 - The shear stresses, in kilopascals (kPa), of nine...Ch. 20 - 20.20 A transportation engineering study was...Ch. 20 - The saturation concentration of dissolved oxygen...Ch. 20 - For the data in Table P20.21, use polynomial...Ch. 20 - 20.23 Use multiple linear regression to derive a...Ch. 20 - 20.24 As compared to the models from Probs. 20.22...Ch. 20 - 20.25 In water-resources engineering, the sizing...Ch. 20 - 20.26 The concentration of total phosphorus and...Ch. 20 - 20.27 The vertical stress under the corner of a...Ch. 20 - Three disease-carrying organisms decay...Ch. 20 - 20.29 The mast of a sailboat has a cross-sectional...Ch. 20 - 20.30 Enzymatic reactions are used extensively to...Ch. 20 - 20.31 Environmental engineers dealing with the...Ch. 20 - An environmental engineer has reported the data...Ch. 20 - The following model is frequently used in...Ch. 20 - 20.34 As a member of Engineers Without Borders,...Ch. 20 - 20.35 Perform the same computations as in Sec....Ch. 20 - 20.36 You measure the voltage drop V across a...Ch. 20 - Duplicate the computation for Prob. 20.36, but use...Ch. 20 - The current in a wire is measured with great...Ch. 20 - 20.39 The following data was taken from an...Ch. 20 - It is known that the voltage drop across an...Ch. 20 - Ohms law states that the voltage drop V across an...Ch. 20 - 20.42 Repeat Prob. 20.41 but determine the...Ch. 20 - 20.43 An experiment is performed to determine the...Ch. 20 - Bessel functions often arise in advanced...Ch. 20 - 20.45 The population of a small community on the...Ch. 20 - Based on Table 20.4, use linear and quadratic...Ch. 20 - 20.47 Reproduce Sec. 20.4, but develop an equation...Ch. 20 - 20.48 Dynamic viscosity of water is related to...Ch. 20 - 20.49 Hooke’s law, which holds when a spring is...Ch. 20 - 20.50 Repeat Prob. 20.49 but fit a power curve to...Ch. 20 - The distance required to stop an automobile...Ch. 20 - An experiment is performed to define the...Ch. 20 - The acceleration due to gravity at an altitude y...Ch. 20 - The creep rate is the time rate at which strain...Ch. 20 - 20.55 It is a common practice when examining a...Ch. 20 - The relationship between stress and the shear...Ch. 20 - The velocity u of air flowing past a flat surface...Ch. 20 - 20.58 Andrade’s equation has been proposed as a...Ch. 20 - Develop equations to fit the ideal specific heats...Ch. 20 - 20.60 Temperatures are measured at various points...Ch. 20 - 20.61 The data below were obtained from a creep...
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