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*launch*### Numerical Methods for Engineers7th EditionSteven C. Chapra Dr., Raymond P. Canale Publisher: McGraw-Hill EducationISBN: 9780073397924

Numerical Methods for Engineers

## Solutions for Numerical Methods for Engineers

Problem 2P:

1.2 Repeat Example 1.2. Compute the velocity to s, with a step size of (a) 1 and (b) 0.5 s. Can you...

Problem 3P:

Rather than the linear relationship of Eq. (1.7), you might choose to model the upward force on the...

Problem 4P:

For the free-falling parachutist with linear drag, assume a first jumper is 70 kg and has a drag...

Problem 5P:

Compute the velocity of a free-falling parachutist using Eulers method for the case where m=80kg and...

Problem 7P:

The amount of a uniformly distributed radioactive contaminant contained in a closed reactor is...

Problem 8P:

A group of 35 students attend a class in a room that measures 11 m by 8 m by 3 m. Each student takes...

Problem 9P:

A storage tank contains a liquid at depth y, where y=0 when the tank is half full. Liquid is...

Problem 10P:

For the same storage tank described in Prob. 1.9, suppose that the outflow is not constant but...

Problem 11P:

1.11 Apply the conservation of volume (see Prob. 1.9) to simulate the level of liquid in a conical...

Problem 12P:

In our example of the free-falling parachutist, weassumed that the acceleration due to gravity was a...

Problem 13P:

1.13 Suppose that a spherical droplet of liquid evaporates at a rate that is proportional to its...

Problem 14P:

Newtons law of cooling says that the temperature of a body changes at a rate proportional to the...

Problem 15P:

As depicted in Fig. P1.15, an RLC circuit consists of three elements: a resistor (R), and inductor...

Problem 16P:

1.16 Cancer cells grow exponentially with a doubling time of 20 h when they have an unlimited...

Problem 17P:

1.17 A fluid is pumped into the network shown in Fig. P1.17. If, and, determine the other...

Problem 18P:

The velocity is equal to the rate of change of distance x(m), dxdt=v(t) (P1.18) (a) Substitute Eq....

Problem 19P:

You are working as a crime-scene investigator and must predict the temperature of a homicide victim...

Problem 21P:

1.21 As noted in Prob. 1.3, drag is more accurately represented as depending on the square of...

Problem 22P:

1.22 As depicted in Fig. P1.22, a spherical particle settling through a quiescent luid is subject to...

Problem 23P:

As described in Prob. 1.22, in addition to the downward force of gravity (weight) and drag, an...

Problem 24P:

As depicted in Fig. P1.24, the downward deflection y(m) of a cantilever beam with a uniform load w...

## Browse All Chapters of This Textbook

Chapter 1 - Mathematical Modeling And Engineering Problem SolvingChapter 2 - Programming And SoftwareChapter 3 - Approximations And Round-off ErrorsChapter 4 - Truncation Errors And The Taylor SeriesChapter 5 - Bracketing MethodsChapter 6 - Open MethodsChapter 7 - Roots Of PolynomialsChapter 8 - Case Studies: Roots Of EquationsChapter 9 - Gauss EliminationChapter 10 - Lu Decomposition And Matrix Inversion

Chapter 11 - Special Matrices And Gauss-seidelChapter 12 - Case Studies: Linear Algebraic EquationsChapter 13 - One-dimensional Unconstrained OptimizationChapter 14 - Multidimensional Unconstrained OptimizationChapter 15 - Constrained OptimizationChapter 16 - Case Studies: OptimizationChapter 17 - Least-squares RegressionChapter 18 - InterpolationChapter 19 - Fourier ApproximationChapter 20 - Case Studies: Curve FittingChapter 21 - Newton-cotes Integration FormulasChapter 22 - Integration Of EquationsChapter 23 - Numerical DifferentiationChapter 24 - Case Studies: Numerical Integration And DifferentiationChapter 25 - Runge-kutta MethodsChapter 26 - Stiffness And Multistep MethodsChapter 27 - Boundary-value And Eigenvalue ProblemsChapter 28 - Case Studies: Ordinary Differential EquationsChapter 29 - Finite Difference: Elliptic EquationsChapter 30 - Finite Difference: Parabolic EquationsChapter 31 - Finite-element MethodChapter 32 - Case Studies: Partial Differential Equations

### Book Details

This is the seventh edition of Chapra and Canale's Numerical Methods for Engineers that retains the instructional techniques that have made the text so successful. Chapra and Canale's unique approach opens each part of the text with sections called "Motivation," "Mathematical Background," and "Orientation." Each part closes with an "Epilogue" containing "Trade-Offs," "Important Relationships and Formulas," and "Advanced Methods and Additional References." Much more than a summary, the Epilogue deepens understanding of what has been learned and provides a peek into more advanced methods. Helpful separate Appendices. "Getting Started with MATLAB" and "Getting Started with Mathcad" which make excellent references. Numerous new or revised problems are drawn from actual engineering practice. The expanded breadth of engineering disciplines covered is especially evident in these exercises, which now cover such areas as biotechnology and biomedical engineering. Excellent new examples and case studies span all areas of engineering giving students a broad exposure to various fields in engineering.

## Sample Solutions for this Textbook

We offer sample solutions for Numerical Methods for Engineers homework problems. See examples below:

Given Information: The equation, dvdt=gâˆ’cmv. The initial velocity v(0) is non-zero. Formula used:...In our example of the free-falling parachutist, weassumed that the acceleration due to gravity was a...1.22 As depicted in Fig. P1.22, a spherical particle settling through a quiescent luid is subject to...Given: The flow chart is as follows: The flow chart, The pseudo code to the provided question is as...2.14 Two distances are required to specify the location of a point relative to an origin in...2.18 Piecewise functions are sometimes useful when the relation-ship between a dependent and an...Given: The number in base-2 is 101101. Formula used: Conversion formula from base-2 to base-10,...3.11 Use 5-digit arithmetic with chopping to determine the roots of the following equation with Eqs....Given Information: The Maclaurin series of exponential function ex=1+x+x22+x33!+â‹¯+xnn!. Formula...

4.6 Use zero- through fourth-order Taylor series expansions to predict x using a base point at....Given Information: The equation f(x)=âˆ’0.5x2+2.5x+4.5. Calculation: The graph of the function can be...5.3 Determine the real root of:
(a) Graphically.
(b) Using bisection to determine the root to....Determine the roots of f(x)=1221x+18x22.75x3 graphically. In addition, determine the first root of...Given: The function, f(x)=sin(x)âˆ’x. The initial condition, x0=0.5 and iterate until Îµaâ‰¤0.01%....6.2 Determine the highest real root of
(a) Graphically.
(b) Fixed-point iteration method...6.7 Locate the first positive root of
Where x is in radians. Use four iterations of the secant...6.23 Determine the roots of the simultaneous nonlinear equations
Use a graphical approach to...Given Information: Provided Polynomial is f(x)=x4âˆ’7.5x3+14.5x2+3xâˆ’20 and the monomial factor is xâˆ’2....Given Information: The given equation is, f(x)=x3+x2âˆ’4xâˆ’4 Use MÃ¼llerâ€™s method. Formula used: The...Use the program developed in Prob. 7.8 to determine the roots of the equations in Prob. 7.5.Given Information: For ethyl alcohol a=12.02,b=0.08407 and the temperature is T=375 K, the number of...Many fields of engineering require accurate population estimates. For example, transportation...8.44 Figure P8.44 shows three reservoirs connected by circular pipes. The pipes, which are made of...8.46 Repeat Prob. 8.45, but incorporate the fact that the friction factor can be computed with the...8.48 Determining the velocity of particles settling through fluids is of great importance of many...Given: The set of equations, 8=6x3+2x22âˆ’x1=x35x2+8x1=13 Consider the set of equations,...A number of matrices are defined as [A]=[415726][B]=[437127204][C]={361}[D]=94213675...9.21 Recall from Sec. 8.2 that determining the chemistry of water exposed to atmospheric can be...Given: The equations, [L][U]=[A] and [L]{D}={B}. Proof: Consider a linear equation represented in...10.3 (a) Solve the following system of equations by LU decomposition without pivoting
(b) ...The following system of equations is designed to determine concentrations (thecsing/m3) in a series...Given: A tridiagonal system: 0.8âˆ’0.40âˆ’0.40.8âˆ’0.40âˆ’0.40.8{x1x2x3}={4125105} Formula used: (a) The...Use the Gauss-Seidel method (a) without relaxation and (b) with relaxation (=0.95) to solve the...11.13 Use the Gauss-Seidel method (a) without relaxation and (b) with relaxation to solve the...Given Information: Flow rates are provided as follows: Q01=6 Q15=3, Q12=4, Q31=1, Q03=8, Q25=1,...Chemical/Bio Engineering
12.10 An irreversible, first-order reaction takes place in four well mixed...Chemical/Bio Engineering
12.12 Figure P12.12 depicts a chemical exchange process consisting of a...Given Information: The air flow through the room with proper description is provided as follows:...Given Information: The function f(x) is given as, f(x)=âˆ’x2+8xâˆ’12 Calculation: Evaluate the first...Employ the following methods to find the maximum of f(x)=4x1.8x3+1.2x30.3x4 (a) Golden-section...Employ the following methods to find the maximum of the function from Prob. 13.7: (a) Golden-section...Given: The function f(x,y)=x2+2y2 at x=2 and y=2 in the direction of h=2i+3j. Formula used: The...14.5 Find the gradient vector and Hessian matrix for each of the following...Given Information: Product A and B are produced during a 40-hr of work week. Company has access to...Given Information: The factors are given as below, Resource Regular Premium Supreme Resource...Given Information: The volume of the cylinder is 0.5 m3. Formula used: Write the expression of...Given Information: Following data is given for a chemical plant that makes three products on a...Given Information: The Manning equation of flow Q (m3/s) can be written as Q=1nAcR2/3S1/2 where n is...Given Information: The parameters are given as Youngâ€™s modulus E is 2Ã—1011Pa, Cross-sectional area A...Given Information: The data is, 8.8 9.5 9.8 9.4 10 9.4 10.1 9.2 11.3 9.4 10 10.4 7.9 10.4 9.8 9.8...Given these data 29.65 28.55 28.65 30.15 29.35 29.75 29.25 30.65 28.15 29.85 29.05 30.25 30.85 28.75...Given Information: The data is, x 0 2 4 6 9 11 12 15 17 19 y 5 6 7 6 9 8 7 10 12 12 Calculation: The...Given Information: The values, log8=0.9030900log12=1.0791812 Formula used: Linear interpolation...Use Newtons interpolating polynomial to determine y at x=8 to the best possible accuracy. Compute...18.14 Develop cubic splines for the data in Prob. 18.6 and (a) predict f (4) and f (2.5) and (b)...Given: The average value formula is f(x)Â¯=âˆ«0xf(x)dxx and the results are, âˆ‘sin(Ï‰0t)N=0,...Given Information: The sawtooth wave shown as, Formula used: Consider f(t) is a periodic function...Given Information: The sawtooth wave given in the following figure, Formula used: Consider f(t) is a...Given Information: Data for specific growth rate k and available food f is given in the form of a...Given Information: The table is given as, Depth, m 0 0.5 1.0 1.5 2.0 2.5 3.0 Temperature, Celsius 70...Given Information: Consider the experimentally measured values of Î³Ë™ and Ï„ from a single blood...20.16 Soft tissue follows an exponential deformation behavior in uniaxial tension while it is in the...Given Information: The provided data is, i âˆ’2 âˆ’1 âˆ’0.5 0.5 1 2 V âˆ’637 âˆ’96.5 âˆ’20.5 20.5 96.5 637...Given Information: The data is, T 0 5 10 20 30 40 Î¼ 1.787 1.519 1.307 1.002 0.7975 0.6529 Graph: The...Given Information: The data is provided as, x=0 x=2 x=4 x=6 x=8 y=0 100.00 90.00 80.00 70.00 60.00...Given: The integral, âˆ«0Ï€2(6+3cosx)dx Formula used: âˆ«abf(x)dx=[F(x)]ab=F(b)âˆ’F(a) Here, F(x) is...21.2 Evaluate the following integral:
(a) Analytically;
(b) Single application of the...Given: Integral, âˆ«03xe2x dx. Formula used: (1) Formula for h is given by, h=bâˆ’an (2) Formula for...Given Information: The given integral is, âˆ«2âˆž1x(x+2)dx Formula used: Simpsonâ€™s 1/3 rule. I1=h3[...Given information: Function, y=cosx Step size, h=Ï€12 The initial value of x, x=Ï€4 Formula used:...Compute the first-order central difference approximations of O(h4) for each of the following...Use a Taylor series expansion to derive a centered finite-difference approximation to the third...Given Information: Write the value of the provided temperatures. T1=âˆ’150Â°C and T2=100Â°C Write the...Given Information: The given expressions are as follows,...24.36 Repeat Prob. 24.35, but use (a) Simpsonâ€™s rule, (b) Romberg integration to, and (c) Gauss...Given Information: The position of a Jet fighter on an aircraft carrierâ€™s runway is, t,s 0 0.52 1.04...Given Information: The vertical velocity of the rocket is,...Given Information: The formula for the upward velocity of rocket is given as, v=uln(m0m0âˆ’qt)âˆ’gt...Given Information: The initial value problem dydt=yt2âˆ’1.1y where y(0)=1. Formula used: Tosolve an...Solve the following problem over the interval from x=0to1 using a step size of 0.25 where y(0)=1....Given Information: Differential equation, dydx=âˆ’200,000y+200,000eâˆ’xâˆ’eâˆ’x . Formula used: The...Given Information: Differential equation, dydt=âˆ’2yt, t=4 to t=5 with step size of 0.5 and initial...Given Information: A steady-state heat balance for a rod is represented as d2Tdx2âˆ’0.15T=0 where the...27.27 The following nonlinear, parasitic ODE was suggested by Hornbeck (1975):
If the initial...A heated rod with a uniform heat source can be modeled with the Poisson equation, d2Tdx2=f(x) Given...27.29 Repeat Prob. 27.28, but for the following heat source: .
Given Information: The differential equation for mass balance of single reactor is, Vdcdt=Qcinâˆ’Qc....Given information: The system of equations, dcadt=âˆ’10cacc+cbdcbdt=10caccâˆ’cbdccdt=âˆ’10cacc+cbâˆ’2cc...Given Information: The series of first order, liquid phase reactions is, Aâ†’k1Bâ†’k2C The second order...Perform the same computation for the Lorenz equations in Sec. 28.2, but use (a) Eulers method, (b)...28.27 The basic differential equation of the elastic curve for a uniformly loaded beam (Fig. P28.27)...Given Information: The dynamic of a forced spring-mass-damper system is given as,...Given information: The upper boundary condition increased to 150Â°C and the left boundary insulated...Given Information: The fixed temperature at top and right boundary is 100 Â°C and the fixed...Given Information: The expression of the temperature distribution of long, thin rod is, l=10...Repeat Example 30.3, but for x=1cm.Given Information: Poissonâ€™s equation: d2Tdx2=âˆ’f(x) Here, function f(x) is the heat source along the...Given Information: To develop nodal equations for the temperature and their gradients at each of the...Given Information: The distributed-parameter system depicts chemical modelling being subjected to...Given Information: Write the expression for Dâ€™Arcyâ€™s law. qn=âˆ’Kdhdn Here, K is the hydraulic...

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