Mathematical Problem Set

Newton Raphson Method

Q1. Use the Newton Raphson Method to determine √7 3 correct to 7 decimal places by considering the function ff(xx)= xx3 − 7.  

Q2 The Colebrook equation is used for turbulent flows through rough pipes. The Colebrook equation is

where f is the Darcy friction factor; ε is the roughness height; Rh is the hydraulic radius and Re is the Reynolds number.

Use the following values: Re = 4900, ε = 0.0045 and Rh = 0.15.

(i) Form an appropriate function g (f) to be used in the Newton Raphson method and plot it.

(ii) Use your graph to determine an initial approximation to the friction factor

(iii) Use the Newton Raphson method to determine the friction factor correct to 7 significant figures.  

Q3 Calculate the determinant to determine which of the following equations have unique solutions.

(i) 5 xx + 3 yy= 2
2 yy− 4 xx= 3

(ii) 2xx−yy− zz= 2
3xx− 2 yy+ 2 zz= −2
− xx+ yy− 3 zz= 3

(iii) 4 xx− 2 yy + 3 zz= 3
2 xx− yy+ zz= −2
2 xx− 3 yy+ 2 zz= −1

Use the inverse matrix to solve the equations where possible. [10 marks]

Q4 Find the 2 values of λ so that the following equations do not have a unique solution:

(2 − lamda) xx− 4 yy= 0
− xx + (3 − lamda) yy= 0