Open University TMA MST121 04 Solutions

TMA MST12I. 04 Cut-off date 21 September 2011
This assignment covers the whole module.

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You should affiwer ALL FOUR questions. One of the requirements for
passing the module is that you must obtain a score of at least 30 out
of 100 for this assignment.
Questionl - 25marks
On 1 September 201.1, a fictitious lake in Norway has a population
of 500 000 adult cra1fish, increasing at a rate of approximately ?.4% per
year without human intervention. The lake can sustain a population of
700000 adult crayfish. It is proposed that each summer 40000 adult
crayfish should be harvested. Let 4,, denote the size of the population
on 1 September r? years after 20L1.
(a) It is suggested that the effect on the population size can be modelled
by the recurrence system
"4o:500000, An+!:L.074An-40000 (n:0,1,2,...).
(i) Briefly explain how this recrurence system is obtained from the
information given at the beginning of the question.
(ii) Use the recurrence system to find the population size predicted
by the model for 1 September 2013 (that is, two yea,rs after 2011).
(iii) Calculate the equilibrium population size according to the model.
(iv) Find a closed form for the sequence A,, given by the model, and
hence find the population size predicted for 1 September 2016.
What is the behaviour of the sequence An for large n?
(v) Use logarithms to find the year when the adult cayfish
population is predicted to be extinct by 1 September, according
to the model, if the level of harvesting remains unchanged.
(b) Adult crayfish (4 years or older) form a subpopulation ofthe crayfish
in the lake, which include hatchlings (under 1 year) and juveniles
(1-3 years). Let Hn and ,I,, denote the sizes of the subpopulations of
hatchlings and juveniles, respectivelS on L September n years af,ter
2011. The following table presents the data for the three
subpopulations.
llatchlings Juveniles Adults
Subpopulation on 30000000 400000 500000
1 September 2011
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141
Birth rate
Death rate
Proposed catch
070
0.31 0.17
0 40000
0
0.99
0
The network diagra,m belosr illustrates the interdependency of the
three subpopulations in the absence of ha,rvesting. It assumes that in
each year all of the surviving hatchlings become juveniles, and that a
third of surviving juveniles become adults.
{r+t An+l
Explain clearly grving both calculations and the reasoning
leading to the calculations, how each of the numbers 0.01, 0.46
and 0.23 in the network diagra^m are derived from the description
of the model and the data in the table.
The recurrence system below is obtained by converting the
network diagram into a matrix model, taking into account the
initiat data for 2011 and the efiect of the proposed annual catch.
It is suggested as a,n alternative model to that in part (a) for the
effect of the proposed annual catch on the number of adult
crayfish.
0
0.46
0.23
Calculate the adult crayfish subpopulation size predicted by the
model for 1 September 20L2.
(iii) According to Mathcad, the model in part (b) predicts that the
subpopulation size of adult crayfish in 2031 will be 863666, which
is greater than the ma:rimum sustainable population of the lake.
Give a reason why the model in part (b) does not give an accurate
prediction for the adult crayfish subpopulation size in 2031.

Question 2 25 marks
This question requircs the use of Mathcail throughout, and no marks will
be awarded to answers obtained by other rn@,ns. For mch of parts (a)-(")
you shoulil pro$,ile an appropri,ate printout, though a printout on one page
mey cnuer your ansuers to seoeral parts. Annotate gour printouts with
Mathmd, tert or handwriting, or reterenen them fram a separate sheet, in
order to etylain clearlg whot you haae done anil what your conclusions are.
You mag find it useful in all parts of this question to refer to A Guide to
Mathcad.
A flexible wire PQRS, of total length 12 metres, is bent into a three-edged
planar shape, and its ends P, ,9 are placed against a disc of
radius I metres with centre O, as shou'n in the diagra,m below. (The arc
PS is not part of the wire.) The end-segments PQ and "E^9 of the wire lie
along straight lines through 0, while the arc QB forms part of a circle
with centre O and subtends an angle r (in radia"ns) at O.
This question concerns the area / enclosed between the wire and the edge
of the disc, which is shoqrn shaded below. This area can be expressed by A: f (x), where
f(r): 9x(4-3r)(16+3c)
2(2 + a)2
(You arc, NOT askeil to show this-)
In part (a), you maE either use the Mathcod gra,ph plotter fiIe (121/,3-Al)
or plot the graph in a new worksheet ol your own.
(a) (i) Plot the graph of the function /(r). Your graph should cover the
interval [0,1.33] in the c-direction a,nd [0,20] in the grdirection.
(ii) By using the 'Tbace' facility (a^nd also, if you wish, the 'Zeom'
fecilif), estimate to two decimal places the coordinates of the
point on this graph at which A : f @) takes its ma:rimum value.
(iii) On the sa,me Sraph, plot the line y:8. Using the 'lbace'facility,
estimate to two decimal places both solutions of the equation
f (s) :8. (These solutions give the values of r for which the
shaded area is 8*2.)
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12)
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(b) Ea€h of the following recrurence relations has the property that if a
sequence generated by the recurrence relation converges to a limit in
the interval [O, f], then that limit must be a solution of the equatiou
/(o) : 8. (You arc NOT asked to show this.)
L6(2 + u)2 A: Sn*1:
B: anr1:
9(4-3r"X16+30*)
8
(n : 0, L,2,. . .)
(n : 0, L,2,. . .)
32 - 3\n6=w
(i) For earh of these recturenc€ relations, generate the sequence with
starting value cs : 1, a,nd tabulate your reeults to six decimal
places. Which sequence converges more rapidly? (That is, which
sequence gives an estimate with specified accuracy for its limit
with the smaller value of rz?)
(ii) Use your tabulated results to write dourn the solutions of the
equation.f(r) :8 to six decimal places.
(c) This part of the question concerns finding the ma:rimum rnlue of the
function /("), * estimated in part (a)(ii), and hence the maximum
possible value of the shaded area .4.
You mag need to put r:: u in gour worksheet before answeri,ng
part (c)(i).
(t Use symbolic differentiation and the 'simplify' keyword to find an
expression for the derivative f'(*).
(ii) The maximum value of /(r) occurs where .f'(r) :0. Use a Bolve
block to solve the equation ft(*):0 for o, grving your answer to
six decima,l places.
(iii) Calculate the corresponding maximum possible value of the
atea A, grving your enswer to four decimal places.
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121
Question 3 25 marks
As for other questions, rernember to show your worlci,ng ecptieitly
thrughout your orutuer to this question.
(a) (i) Use the Composite Rule to differentiate the function
'f(t): st'+"*'' t5l
(ii) Use the Product Rule and your answer to part (a)(i) to show that

(iii) Find any stationary points of the function g(c) defined in
part (a)(ii), and use the First Derivative Test to classify each
stationary point as a local maximum or a local minimum of g(n).
(Note that the domain of the firnction g is the interval [0,22r], and
recall that lcosrl ( 1 for all values of o.) t5l
(iv) Using your answers to parts (aXii) and (a)(iii), find the area
below the graph of
a:(l-2cosr)(3*2cosr) sia+coon (0 < r <2n)
and above the s-a:ris. Give your a.nswer to five significant figures. [41
(b) (t Ueing your answer to part (*Xi), find the general solution of the
difierential equation

tion in implicit form. I4l
(ii) Find the particula,r solution of the differential equation in
part (bXi) for which U : -2 when c : 0, a,nd then give this
pa,rticular solution in explicit form. t3l
Question 4 25 marks
This question rcquires the use of OUStats.
(') The file AGES.OUS contains data on the ages in months of 113
students enrolled on an elementary statistics course. This part of the
question concerns these data. [Lindgren, B. W. and Berry, D. A.
(1981) Elementary Statistics, Macmillan, page 67.]
(0 Use OUStats to find the age of the youngest student enrolled on
the course. What facility of OUStats did you use to find this
value?
(ii) Obtain a printout of. a histogram of the data using starting value
210 months and an interval width 50 months. State, with a brief
explanation, & reason why a normal distribution might not be a
suitable model for the variation of these data.
(iii) Obtain a printout af. a frequencg diagmm of the data using
starbing value 200.5 months and interval width 20 months.
(iv) Making use of the frequency diagram obtained in part (aXiii),
write dovrn the most popular age interval for the enrolled
students. How many enrolled students are in this age interval?
Find the number of enrolled students who are 25 years old exactly
or less.
The file ALLOYS.OUS contains data related to thirteen specimens of
cupro.nickel alloys, which were tested for corrosion in salt water for
60 days. 'Iron' contairu the iron content of each specimen (the units
are not given in the source) and 'Loss'contains the corresponding
weight loss in mg per squsre decimetre per day. The data are paired.
[Draper, N. R. and Smith, W. L. (1966) Applieil Regression Analysis,
John Wiley, page 37.1
(i) Obtain a printout of a scatterplot of 'Iron' and 'Loss', using 'Iron'
as the explanatory variable (which must be on the c-axis). Add
the least squares fit line to the plot.
(ii) Describe any pattern that you observe on the plot.
(iii) Obtain the equation of the least squares fit line of 'Loss' on 'Iron'.
If you use labels other than 'Iron' and 'Loss', specify which
variable represents 'Iron' and which represents 'Loss'.
(iv) Using the model from part (bxiit, estimate the mean weight locs
due to corrosion for a cupro-niclcel alloy with iron content of 0.1.
(v) The first observation has iron content of 0.01 and weight loss due
to corrosion of L27.6 mg per square decimetre per day. Using the
model from part (bxiii), find the residual for this observation.
(vi) Briefly explain why it would be inappropriate to use the model
from part (bxiit to estimate the mean weight loss due to
corrosion for a cupro-nickel alloy with iron content of 2.3.
T4
[4)
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(b)
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121
121
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121
CMA MSTl2I. 42
This CMA contains 28 questions.
Cut-off date 28 September 2011
Questions 1 to 7 are on Block A.
Question L
Choose the TWO options that are false statements.
Options

Question 3
Choose the option that gives the equation of the line that pa.sses through
the point (-2,4) and is perpendicular to the line 3c - U:5.
Optians
A 3c-g:-10 B f*39:16 C 3r-g:5
D 3*ay:-2 E r-3g:5 F 3r*g:l$
G s*39:-5 H r-3U:-10
10
Question 4
The method of completing the square can be used to write the expression
5n2 - 6n * 7 in the form a(r+ b)2 + c, where o, b and c are constants.
Choose the option that gives the value of c.
Options
A-t B-2 c# DT Er# F
Question 5
Choqse the option that gives the domain of the function /(o):#
Options
A, [-1,4) B (-oo,4l C (-4,1)
D (-1,4) E R F [-1,m)
Question 6
Choose the option that is a false statement.
Options

Question 12
Iltug forces (and no others) act upon an object, which remains at rest.
The forces are represented by the vectors p,-e and R, where
P:5i*4j and Q:-ii+6j.
choose the TWo options that give the magnitude and direction of the
vector R,, to one decimal place.
Options
A Magnitude 3.7
C Direction -168.2'
E Direction -78.7o
G Direction 11.3o
B Magnitude L0.2
D Direction -101.3o
F Direction -11.3o
H Direction 78.?o
Question 13
The vector a has magnitude 2.4 and direction -Llzo, and the vector b has
magnitude 2-1 and direction -261 choose the option that gives the
component form of the vector 2a - lb, to one decimal place.
Options
A -11.2i - 11.0j
D 0.2i - 11.0j
G 10.4i - 12.8j
B -8.4i - 4.1j
E 3.6i + S.2j
H 11.2i + 11.0j
c -7.3i - 12.8j
F 7.3i + 4.1j
13
Question 14
With respect to the usual conveation for labelling a triangle (shown in
Figure 3.1(a) on page 30 of Chapter B3), the triangle .4BC has side
lengths o : I and b : 5, and angle A: 42". Choose the option that gives
the angle B of the triangle, to one decimal place.
Options
A 4.30 B L7.70 C 2l.go
D 158.20 E 162.30 F L75.7"
G It could be either 4.3o or 175.70.
H It could be either 21.8aor 158.2o-
Questions 15 to 21 a,re on Block C.
Qumtion 17
By first applying the Quotient Rule, choose the option that gives (correct
to two decimal places) the overall manimum value of the function
r e^2
f (a): r;#
on the interval [1,7].
Options
A 1.88 B 2.10 C 4.15 D 4.88
E 5.09 F 5.32 G 6.73 H 7.51
Questions t5 and 16
The function / is given by
f(s): ln(e'+ e-'+2).

Quetions 24 and 25
The distribution of the weighte of the contents of packs of tea bags,
labelled as @ntafuing 125 g, has mean 126.1 g, and standaxd
deviation 3.2g.
24 Choose the option that is closet to the standard ermr of the mean
weight (in gra,ms) of the contents for sa,mples of 50 packs.
Options for Question 2l
A 0.06 B 0.45 C 0.51
D 0.61 E 0.79 F L.7s
You will neeil to we OUStats to answer Question 25.
25 The mean weigbt (in gra.rrs) of tbe contents for samples of 30 packs of
the same tea bagu has standard error 0.58 g. Chome tbe option that
is closest to the probability that the mean weight of the contents for a
sa,rnple of 30 packs of tea bags is betveen 124.5 S and 125.5 g.
Options for Qucstion 25
a 0.15 B 0.20 c 0.25
D 0.30 E 0.75 F 0.85
t7
Question 7
A popula.r pastime in a particular club is flipping beer mats balanced on
the edge of a table and trying to catch them before they land back on the
table. Ann is successful two times out of five in the long nrn. Choose the
option that is closest to the number of times, on aver&ge, that she has to
flip a beer mat ia order to catch it.
Options
A'0.4 80.5 Cl DL.2 E2 F2.5
Question 8
A game uses a pack of five cards numbered I,2,3,4, 5. In each round the
cards are shufred, the top card is turned face up, and the number is
recorded. This card is then returned to the pack. Choose the option that
gives, on &verage, the number of rounds to two decimal places that would
be required to obtain four different numbers.
Options
A 5.42 B 6.42 C 7.42
D8 E 8.42 F 10.42
Question 9
Assuming that a birth is equally likely to occur otr any day of the week,
choose the option that is closest to the probability that, in a fa,urily of four
children, at least two of the children were born on the sa,me day of the
week.
Options
A 0.02 B 0.10 c 0.35
D 0.45 E 0.55 F 0.65
Questi,ons 10 to 13 are on Cha,pter D2.
Question 10
A sample of concentrations of transferrin receptor for six women with
laboratory evidence of overt iron-deficiency anaemia yielded the following
data, in ppm.
L2.1 5.3 7.7 5.4 5.9 13.2
Choooe the option that is closest to the sa,mple standard deviation.
Options
A 3.19 B 3.52 C 8.30
D L2.23 E 24.96 F 27.34
18
Questions 26 and' 27
A study examined the durability of two brands of tyres by using a mactrine
with a metallic device that wore dourn the tyres. The time it took (in
hours) for eanh tyre to blow out was recorded. A zummary of the data is
given in the table below.
Sa'mple siae Sample mean Ssmple standaxd deviation
Bra"nd 1 15 3.84
Brand 2 18 4.24
0.59
0.39
2B Choose the option that gives the estimated standard error (ESE) for
the difference between the two population mea,llsl' correct to two
decimal places.
Options for Questi,on 26
A 0.03 B 0.13 c 0.18
D 0.25 E 0.28 F 0.51
27 The twesaurple z-test is to be gsed to determine whether there is a
difference, at the 5% significance level, between the mean blorn'-out
times of the two brands of tyres. choose the TWO options that give
the maguitude of the tmt statistic, to two decimal places, and the
conclusion obtained from the test.
Options lor Question 27
A 0.01 B 0.02 c 1-10
D 1.20 E 2.00 F 2-02
G There seelns to be a significant diference between the mean blow-out
times of Brand L and Brand 2 tyres, at the 5% significance level'
H There seeuls to be no siguificant diference between the mean blo*out
times of Brand I and Brand 2 tyres, at the 5% significance level.
18
Questions Ll to 13
Yau should use OUStats for these questions.
The weights (in ke) of people in a sarnple selected at random from a
particular population are normally distributed with mean 73.15 and
standard deviation 6.6.
11 Choose the option that is closest to the weight above which
approximately 25To of the weights of people from the population will
lie.
Options for Question 11
.4. 55.20 B 55.87 C 56.63
D 60.24 E 68.70 F 77.60
12 Choose the option that is closest to a range of values, symmetric
about the mean, within which approximately 95% of the weights of
people will lie.
Options for Question 12
A (60.2,86.1) B (60.6,85.7) C (62.3,84.0)
D (66.9,88,4) p (62.6,83.?) F (61.6,84.7)
13 Choose the option that is closest to the percentage of people with
weights below 90kg.
Options for Question 13
AL% BL0% Cts%
D 8L% E 90% r sgTo
Questions 1l to 17 &tt on Ghapter D3.
Questions 14 to 16
The distribution of the weights of the contents of borces of a certain
breakfast cereal, labelled as containing 4509, has mean 452.1g and
standard deviation 6.5 g.
LA Choose the option that is closest to the standard error of the mean
contents (in grams) for samples of 30 cereal boxes.
Options for Questilon 1l
A O.22 B 0.32 C 1.19
D 1.21 E 1.30 r 2.5
19
Question 2E
The comparative effectiveness of conventional and digitat image libraries
was investigated by comparing the times (in seconds) that thirteen
computer-proficient medical professionals took to retrieve the same image
from a library of slides and from a computer database with a web
interface. Bo:cplots for the data are shown below, where ,Slide' and
'Digital'refer to the slide library and to the database, respectively.
Boxploft Slide, Digttal
0204060-l
Values
Choose the option that nrzst be a true statement if the bo:rplots are
correct.
Options
A More data were collected for 'Slide' than for ,Digital,.
B The range for 'Slide' is equal to the range for ,Digital,.
C The interquartile range is greater for 'Slide'than for ,Digital,.
D The mean retrieval time for 'Slide' is about 35 seconds.
E The lower quartile is smaller for 'Slide'than for ,Digital'.
F The upper quartile is smaller for 'Slide' than for ,Digita,l'.
Digital *W-
19
You shoulil use OUStats for Questions 15 anil 16.
The sa,mpling distribution of the mean weight of contents, for sa,mples of
35 cereal boxes, has mean 452.19 and standard deviation 1.1g.
15 Choose the option that is closest to the probabitity that the mean
weight of the contents of a sarnple of 35 cerea,l bores will be less
than 450g.
Options for Question 15
A 0.03 B 0.17
D 0.53 E 0.79
16 Choose the option that gives a range of .ualues, symmetric about the
mean, within which the mean weight of contents of approximately
gOTo of. sa,mples of 35 cereal boxes will lie.
Options for Question 16
A (449.2,455.0) B (449.4,454.3)
D (450.3,453.9) E (451.0,453.2)
c 0.31
F 0.97
c (449.9,454.3)
F (451.4,452.8)
Question 17
The mean monthly expenditure on petrol per household in Milton Keynes
is estimated by selecting a random sa,urple of 36 households. The sample
mean is 1186.25, and the sa,rnple standard deviation is t47.40.
Choose the option that gives an approximate g5% confidence interval for
the mean monthly expenditure on petrol per household in Milton Keynes.
Options
A (r7L.24,20L.26)
D (L7A.77,201.73)
B (173.29,199.21)
E (173.61,198.89)
c (175.19,1e7.31)
F (174.80,197.70)
20
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{
Qaestions 18 to 24 orv on Chapter D4.
Questions lE to 20
The salaxies of employees (in f) in a certain small company are as follows.
9829 12554 30802 23664 17582
4L54L 29263 27486 45859 14291
lE Choose the option that is the media,n salary in t in the company.
Options for Question 18
A 9829 B 23ffi C 25287
D 25575 E 27486 F 45859
19 Choose the TWO options that give the lower and upper quartiles for
the salaries in the company.
Options for Qaestian 19
A 9829 B 14291 C 17582
D 29263 E 30802 F 45859
20 Chooee the option that gives the range of the salsrie in the comparry.
Optians lor Question 20
A 9829 B 18015 C 25287
D 25675 E 36030 F 45859
21
Questions 2L and 22
A small supermarket wishes to test the effectiveness of two types of
coupon for a day. The a,mount (in .e) spent by each customer using a
coupon is recorded.
A summary of the resr:lts is given in the following table.
Amount (i" f) spent during the day
Coupon Sa,mple Sa.mple Sample standard
str,e mean deviation
1}pe I 38 26.27 4.37
Type II 32 24.05 9.4
2L Choose the option that is closest to the estimated standa,rd ermr
(ESE) of the difference between the sample means for the first and
second types of coupon.
Options for Question 21
A 0.11 B 0.64 C 1.81
D 3.26 E 6.02 F 10.37
22 Choose the option that is closest to the magnitude of the test
statistic a that would be used to carry out a two-sample z-test to
determine whether there is a difereuce between the mean amount
spent with the two Wpes of coupon.
Opti,ons for Question 22
A0 80.06 c1.11
D 1.23 E 2.22 F 27.8A
22
Questions 28 and 24
A production manager ia a certain engineering company uses a regression
line to model the relationship between production volume and production
cost. The regression line obtained using the recorded data has the equation
u:4.4*3-2*,
where y is the production cost (in .f000s), and c is the units produced
(in 000s).
28 Choose the option that gives the estimated production coet (in f,000s)
for producing 5500 units.
Options for Question 23
A 17.60 B 22.00 C 27.4A
D 5500 E r7ffi4.40 F 17600004.40
24 Chome the option that gives the estimated inerease in the production
cost (in .0000s) due to a production increase from 40fi) to 5000 units.
Options for Question 2l
A3.2 84.4 C10
D 1000 E 3200 F M00
23

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Subjects: Calculus
Topic: Differential Equations
Level: University
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TMA MST12I. 04

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Open University TMA MST121.pdf

Subjects: Calculus
Topic: Differential Equations
Level: University
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TMA MST12I. 04

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TMA MST12I. 04

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