EBK NUMERICAL METHODS FOR ENGINEERS
7th Edition
ISBN: 9780100254145
Author: Chapra
Publisher: YUZU
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Chapter 15, Problem 6P
Use a software package (for example, Excel, MATLAB, Mathcad) to solve the following constrained nonlinear optimization problem:
Maximize
Subject to
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Verify if the following functions are Linear or not. Support your conclusion with appropriate
reason.
a) F(x) =
b) f(x) =rcos wt
3.1-6
DI 3.1-6. Use the graphical method to solve the problem:
Maximize
Z = 10x + 20x2,
subject to
-X + 2x, s 15
X1 + x2 < 12,
5x, + 3x s 45
and
X 0, x2 2 0.
3.1-11 (a) and (b) only
For (a), please specify: (1) Decision variables; (2) Objective function; (3) Constraints.
MacBook Pro
Use the graphical method to find the optimal solution for the following LP equations:
Min Z=10 X1 + 25 X2
Subject to X1220, X2 ≤40 ,XI +X2 ≥ 50
X1, X2 ≥ 0.
Chapter 15 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
Ch. 15 - A company makes two types of products, A and B....Ch. 15 - Prob. 2PCh. 15 - 15.3 Consider the linear programming...Ch. 15 - 15.4 Consider the linear programming...Ch. 15 - Use a software package (for example, Excel,...Ch. 15 - 15.6 Use a software package (for example, Excel,...Ch. 15 - 15.7 Consider the following constrained nonlinear...Ch. 15 - Use a software package to determine the maximum of...Ch. 15 - Use a software package to determine the maximum of...Ch. 15 - Given the following function,...
Ch. 15 - You are asked to design a covered conical pit to...Ch. 15 - 15.12 An automobile company has two versions of...Ch. 15 - 15.13 Og is the leader of the surprisingly...Ch. 15 - 15.14 Develop an M-file that is expressly designed...Ch. 15 - 15.15 Develop an M-file to locate a minimum with...Ch. 15 - Develop an M-file to implement parabolic...Ch. 15 - 15.17 The length of the longest ladder that can...
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- OPERATIONS RESEARCH solve using the simplex solution (mixed constraints) show all the iteration • show the complete solution ● Convert the LP Model to its standard form: Min Z = 2X, + 3X₂ Subject to: 1/2X₁ + 1/4X₂ X₁ + 3X₂ X₁ + X₂ X₁, X₂ ≤4 ≥ 36 = 10 ≥0arrow_forward3. Using the trial function u¹(x) = a sin(x) and weighting function w¹(x) = b sin(x) find an approximate solution to the following boundary value problems by determining the value of coefficient a. For each one, also find the exact solution using Matlab and plot the exact and approximate solutions. (One point each for: (i) finding a, (ii) finding the exact solution, and (iii) plotting the solution) a. (U₁xx -2 = 0 u(0) = 0 u(1) = 0 b. Modify the trial function and find an approximation for the following boundary value problem. (Hint: you will need to add an extra term to the function to make it satisfy the boundary conditions.) (U₁xx-2 = 0 u(0) = 1 u(1) = 0arrow_forward3. Using the trial function uh(x) = a sin(x) and weighting function wh(x) = b sin(x) find an approximate solution to the following boundary value problems by determining the value of coefficient a. For each one, also find the exact solution using Matlab and plot the exact and approximate solutions. (One point each for: (i) finding a, (ii) finding the exact solution, and (iii) plotting the solution) a. (U₁xx - 2 = 0 u(0) = 0 u(1) = 0 b. Modify the trial function and find an approximation for the following boundary value problem. (Hint: you will need to add an extra term to the function to make it satisfy the boundary conditions.) (U₁xx - 2 = 0 u(0) = 1 u(1) = 0arrow_forward
- Question 4. Consider the instance F4 || Cmax Wwith no buffer (means jobs are not unload from previous machine until next machine is not available) in front of all machines except M1. Apply NEH Algorithm to minimize the Cmax Jobs 1 2 3 6 М1 17 13 29 23 37 31 М2 25 27 33 31 37 41 M 3 52 14 43 34 25 47 М4 28 31 48 43 18 17arrow_forwardThe values of p and h which renders (makes) the following set of equations dynamically and statically decoupled are, respectively. k,+k2 5 p+4 x1 = 0, X2 7 h+1 + [7h+1 J+e 5 p+4 J+e h= -0.214 and p = -1.76 h = -0.143 and p= -0.8 h = -0.281 and p= -1.2 h = -0.081 and p = -0.536arrow_forwardYou are the mechatronics engineer of a manufacturing plant. You decide to perform an analysis on a robot arm of the assembly line with the objective of optimizing its performance. After taking several readings of the speed of the arm’s end effector, you approximate its velocity to the function given below. v(t) = -t4 + 5t3 - 7t2 + 3t + 0.22 0 =< t =< 3 where the velocity is in ms-1 d) Knowing that the distance travelled by an object is the area under its velocity-time graph, determine the distance travelled by the end effector on the interval 0 =< t =< 1 by using the mid-ordinate rule. Simpson’s rule correct to 3 decimal places using four intervals. e) Calculate the same distance as in (d) above by using the appropriate definite integral. f) Compare the distances you calculated in (d) and (e) above and comment on the accuracy of the two methods you used in (d)arrow_forward
- You are the mechatronics engineer of a manufacturing plant. You decide to perform an analysis on a robot arm of the assembly line with the objective of optimizing its performance. After taking several readings of the speed of the arm’s end effector, you approximate its velocity to the function given below. v(t) = -t4 + 5t3 - 7t2 + 3t + 0.22 0 =< t =< 3 where the velocity is in ms-1 Use your knowledge of differentiation to sketch the velocity-time graph, clearly marking the critical points. Using the graph sketched in (a) above, estimate the velocity when t = 1.5 s Calculate the velocity of the function when t = 1.5 s by substituting to the velocity function. Compare this value to the value you estimated in b above.arrow_forwardDo not actually solve the problem numerically or algebraically, just pick the one equation and define the relevant knowns and single unknown. Don’t forget to include direction when called for by a vector variable 12) The air conditioner removes 2.7 kJ of heat from inside a house with 450 m3 of air in it. At a typical air density of 1.3 kg/m3 that means 585 kg of air. If the specific heat of air is 1.01 kJ/(kg oC), by how much would this cool the house if no heat got in through the rest of the house during that time?arrow_forwardDiscuss the process of solving engineering problems using MATLAB • Linear interpolation • Cubic-spline interpolationarrow_forward
- Suppose you want to solve the following LP: maxcTx: Ax≤b, but unfortunately it is infeasible (think e.g. about an inventory problem where the demand cannot be satisfied by the warehouse). Let A ∈ Rm×n. Now suppose you can “augment” your LP by buying some more slack in your problem (think e.g. of buying some of the product from other warehouses). In particular, for i = 1, . . . , m, if you want to increase the right-hand side of the i-th constraint by some value λi, you will pay diλi for some fixed number di. Suppose moreover that the right-hand side of the i-th constraint can be augmented by at most ki, for i = 1,...,m. How can you find the optimal augmentation, i.e. the one that maximizes the profit of the optimal solution of the augmented LP minus the cost for the augmentation?arrow_forwardSolve the initial value problem. y" + 4y' + 20y = 0: y(0) = 2 y (0) = - 3 %3D Chapter 6, Section 6.2, Go Tutorial Problem 12 Find Y(s). 2s + 5 Y(s): s2 + 4s + 20 2s + 5 Y(s) = s2 + 4s + 20 2s – 5 Y(s) : + 4s + 20 2 + 4s + 20 Y(s) 2s + 5arrow_forwardQ1: The number of bacterial cells (P) in a given reactor is related to time in days (t) as described by the following mathematical model: dp dt 0.0000007 P², If at initial time (P = 106). Determine the number of cells when (t 2days) using the fourth order Runge-Kutta method and at time increment of (1 day). = = 0.3 P 1arrow_forward
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