Task 1 (This is a Matlab task) (50 marks)

Suggest an engineering problem that involves a system of linear equations of 4 equations and 4 unknowns which can be modeled as a matrix equation of the form Ax=b. You can choose a problem related to any of the following areas:

•      Linear circuits

•      Statically determinant truss Summing forces in a certain direction and using

Newton’s first law 

•      Dynamics Summing forces in the vertical direction and using Newton’s second law of motion

•      Any other civil or architectural engineering problem of your choosing You are required to do the following:

1.      Within the Matlab script file (using % comments): Explain the engineering problem and how it is modeled as a system of linear equations by defining quantities that the unknown variables represent, convert the resulting system of linear equations to a matrix equation  and choose the parameters of the system (the matrix A and the vector b)(10)

2.      Design a Matlab script file to solve the system using matrix inverse. The Matlab code must do the following:                                                     

a.       Compute the determinant of the matrix using cofactor expansion   (5) 

b.      Compute the inverse of the matrix by computing all the cofactors and the adjoint of the matrix    (20)

c.       Find the solution of the system Ax=b using the matrix inverse  (5)  

3.      Explain each of your Matlab code use comments (starts with %) in the script file (10)

Task 2  ( 50 marks)

The strength of concrete follows a Normal (Gaussian) distribution. The following table shows various concrete grades and their mean and standard deviation compressive strength.

You are required to do the following

1. Select a concrete type from the table below and write a Matlab script to perform the following:                                                                      

a.       Plot the probability density function (PDF) of the random variable that represents the strength of the material you selected.  (5)

b.      Plot the cumulative distribution function (CDF) of the random variable that represents the strength of the material you selected. (5)

c.       Find the probability that the concrete strength is above 30 N/mm²          (10)

d.      Select a range of values of the strength and find the probability that the concrete strength is in that range.   (10)

e.       If the concrete material you selected is mixed with another material such that its strength is transformed according to the linear transformation  = + where s is the strength of the original material, y is the strength of the resulting material, and a and b are constants. Select any two values for a and b and find the probability that the strength of the new material is above the mean value of the original material.  (10)

f.        Explain each of your Matlab code use comments (starts with %) in the script file  (10)