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UNI VERSITY OF MAURITI US

FACULTY OF ENGINEERING

SECOND SEMESTER/YEARLY EXAMI NATI ONS

MAY 2011

PROGRAMME BEng (Hons) Civil Engineering

MODULE NAME Numerical M ethods and Statistics

DATE Tuesday

10 May 2011

MODULE CODE CIVE 2001Y(3)

TIME 13:30 16:30 Hrs DURATION 3 hours

NO. OF

QUESTIONS SET

6 NO. OF QUESTIONS

TO BE ATTEMPTED

5

INSTRUCTIONS T O CANDIDATES

Answer Question 1 which is Compulsory and any other Four (4) questions.

Answer Five (5) questions in all .

List of Formulas and Statistical Tables are attached.

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NUM ERICAL M ETHODS AN D STATISTI CS CIV E 2001Y(3)

Question 1 [COM PULSORY]

(a) The table below gives n = 10 paired measurements of effluent BOD 5 and COD.Interpret the data using correlation.

COD

(mg/ L)

9.7 8.8 8.6 6.3 9.7 15.4 7.6 8.1 7.8 11.1

BOD5 5.0 6.1 5.5 4.2 4.3 4.0 4.4 5.9 3.5 5.4

[8 marks]

(b) Water samples are collected from a residential area that is served by either thecity w ater supp ly or by pr ivate wells. The samples are analysed for thei r

mercury concentrations in g/L. The data recorded are as fol lows.

City Water Supply

Sample Size M ean Variance

13 0.157 0.0071

Pri vate Well s 10 0.151 0.0076

Conduct the appropriate hypothesis testing to determine if the mercury

concentrations are different in the two supplies. [12 marks]

Question 2

(a) A flat plate of mass m falling freely in air with a Velocity V is subject to adownward gravitational force and an upward frictional drag force due to air.

The drag force fd is given by the expression

V02.0

Vln500V3.0 3

2

df

Terminal Velocity is reached when the drag force equals the gravi tational force

f = fd mg = 0

Find the terminal velocity using the bisection method if m = 1 kg and g = 9.81

m/ s2. Use an ini tial in terval of v = 0 to 200m/ s. Show your work for computing

the fi rst 3 iterations of the bisection method. [8 marks]

(b) In the turbulent flow of f lu id i n a smooth pipe, the fr ictional force on the fluid isrepresented in terms of a fr iction factor f , which is posit ive and l ess than 0.1. The

equation for f is 8.0Relog21 10 ff

, where Re is a constant, called the

Reynolds number, which varies with the fluid properties, flow rate and pipe

diameter. Use the Newton Raphson method to obtain an approximate value for

the friction factor (f) if the Re = 104. Show your computations for the fi rst 3

iterations of the Newtons method. [12 marks]

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NUM ERICAL M ETHODS AN D STATISTI CS CIV E 2001Y(3)

Question 3

The data describing the storage volume to surface area relationships for a particular

reservoir i s given in the follow ing table:

Storage Volume (km3) 19.5 14.19 9.71 5.92 3Surface Area (x106m2) 204 241.6 207.4 168.8 124.5

Write the polynomial in Lagrange form that passes through the points, then use it to

estimate the value of the surface area when the storage volume is 11.0 km3.

[20 marks]

Question 4

Suppose you are planning to use a large parabolic arch with a shape given by:

y = 0.1x(30 x)

where y i s the height above the ground and x i s in metres.

Calculate the total length of the arch by using Simpsons rule. Divide the domain f rom

x = 0 to x = 30m into 10 equally spaced intervals. The total l ength of the arch is given

by

L = dxdx

dy2

30

0

1

[20 marks]

Aide Mmoire:

Simpsons3

1 rul e: I(f) =3

h )(24 1

7,5,36,4,2bfxfxfaf

N

j ii

N

i

Simpsons8

3rul e: I(f) =

8

3h )(21(3 2

10,7,4

1

8,5,2 bfxfxfxfafN

j iii

N

i

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NUM ERICAL M ETHODS AN D STATISTI CS CIV E 2001Y(3)

Question 5

Consider the follow ing f irst order ODE

22 xy

dx

dy from x = 1 to x = 2.8 wi th y(1) = 2.

Solve with the classical four th order Runge Kutta method using h = 0.6.

Compare the resul ts wi th the exact (A nalyt ical Solution)

4

1

4

3

2

1

2

1 222 xexxy

Aide Mmoire:

y i+1 = yi + 6

1

(k1 + 2k2+ 2k3 + k4)hk1= f(xi,yi)

k2 = f(xi + h, y i + k1h)

k3 = f(xi + h, y i + k2h)

k4 = f(xi + h, y i + k3h)

[20 marks]

Question 6

Using least squares regression, fit a parabola (second order polymonial) to the

following data:

x 1 2 2.5 4 6 8 8.5

y 0.4 0.7 0.8 1.0 1.2 1.3 1.4

(a) Write the equation for your parabola, stating the unknown coefficients[2 marks]

(b) Write the set equations needed to compute the unknown coefficients using theleast squares method [8 marks]

(c) Solve for the coefficients using Gauss Elimination [10 marks]

END OF QUESTION PAPER

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