Global Population Models and Logarithmic Scales
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Global Population Models and Logarithmic Scales
Part A: Global Population Models
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Question 1 ( KAPS 1.5 )
Major issues associated with planetary population growing:
 Corporate demand put on the finite resources of the Earth by the increasing population
 Population growing is besides known to lend to many environmental issues such as:
 Climate alteration
 Increased glade of rain forests
 Loss of 10s of 1000s of works and wildlife species
 Increased nursery gas emanations
 Development of the Earth’s surface land
 Decreasing biodiversity
 Economic stagnancy in poorer societies due to the inability to maintain up with demand for resources
Reasons for calculating future population figures:
 To estimate future demand for nutrient, H2O, energy and legion services
 To calculate future demographic features
 To alarm policymakers about tendencies that may act upon economic development
 To help in making policies that suit assorted projection scenarios.
Modeling the population growing
Question 2 ( KAPS 1, 1, 2 )
 To make a theoretical account for the population of Australia, the initial population () and the per centum growing per twelvemonth () must be substituted into the undermentioned exponential equation:
Substituting,,
Therefore, the theoretical account derived from equation for the population of Australia is:
The theoretical account for the population of Australia () assumes that the population growing remain changeless overold ages. However, if the population growingdoes non stay changeless, the theoretical account will non be valid as the attendant populationwill be wrong.
 To foretell the population of Australia in 2020,must be substituted into the theoretical account created above for the population of Australia.
Substituting
( nearest whole figure )
Therefore in 6 old ages ( 2020 ) the population will be aboutpeople. This represents an mean addition of aboutpeople per twelvemonth which is sensible as there was an addition of 405 400 people from September 2012 to September 2013 ( Australian Bureau of Statistics, 2014 ) .
[ Check ]
Substituting
Therefore the solution is justified.
 To happen the twelvemonth in which the population exceeds 30 million, the valuemust be found by replacing amillion into,
Substituting
(Converting into log)
( 2dp )
Harmonizing to the theoretical account found above, The population of Australia will transcendmillion in 14 old ages 2 months ( rounded up to 2 months because it must transcend 30 million ) .
old ages, 2 months (nearest valid whole)
Therefore, the population of Australia will transcendinold ages, 2 months ( February of 2028 ) . The mathematical theoretical account used in the inquiry is valid whenand. It besides assumes a changeless growing rate which will ensue in an inaccurate projection if the premise made is found to be wrong harmonizing to the scenario.
[ Check ]
Substitutinginto
( 2dp )
Although whenis substituted back into the inquiry, the right manus side is non equal to the left manus side, this is due to replacing a rounded value forT. Therefore, the solution is justified as the value when substituted gives a value which is merely 0.01 % ( 2dp ) more than 30 million
Question 3 ( KAPS 3 )
To happen the approximative growing rate of planetary population assumed by the predictor, the current population, the estimated population in 2050and the clip in old ages sincedemand to be substituted into the population exponential equation:
Substituting
( 8dp )
6822 ( 1 ) ( 6dp )
( Converting into % ) ( 2dp )
Therefore, the predictor assumed a planetary population growing rate of approximative 0.68 % ( 2dp ) when geting at planetary population of 9 billion in 2050 presuming that the growing rate is changeless overTold ages.
[ Check ]
Substitutinginto,
( 2dp )
Although when( 0.68 % ) is substituted back into the inquiry, the right manus side is non equal to the left manus side, this is due to replacing a rounded value forT. Therefore, the solution is justified as the value when substituted gives a value which is about merely 0.08 % ( 2dp ) less than 9 billion.
Question 4 ( MAPS )
In order to develop the most accurate population theoretical account utilizing the planetary population informations ( on the left ) , 4 different cardinal theoretical accounts were used. The arrested developments developed included linear, exponential, power and order 2 multinomial ( quadratic ) . Thevalue was displayed on each chart so that the truth of arrested development can be carefully compared to happen the most appropriate theoretical account. Since the most appropriate theoretical account will hold avalue closest to 1, it was found that the order 2 multinomial arrested development best tantrum the planetary population informations. Although threedimensional and quartic arrested developments were found to hold a highervalue, they would non pattern population growing accurately as there would be minimums in certain topographic points which would propose that the population declined as some phase which ( harmonizing to the informations provided ) is wrong.
Comparison ofValuess ( 7dp ) 

Type of Arrested development 
value 
Logarithmic 
0.9938602 
Linear 
0.9944594 
Exponential 
0.9948708 
Power 
0.9954035 
Order 2 Polynomial 
0.9992986 
Table 1 Comparison of R2 Values
Graph 1 Logarithmic Regression Model
Graph 2 Linear Regression Model
Graph 3 Exponential Regression Model
Graph 4 Power arrested developmentGraph 5 Polynomial ( Order 2 ) Arrested development
To measure the anticipation that the planetary population will be 9 billion in 2050 utilizing the multinomial ( order 2 ) theoretical account below,must be substituted into the equation:
…
Substituting,
( nearest valid whole a?µ exponential growing )
Harmonizing the theoretical account developed, the population in 2050 will be about 11,250,594,405. Before measuring the anticipation of a population of 9 billion in 2050, it is necessary to analyze the theoretical account in context. When replacing a twelvemonth that occurs after the information scope ( 19502010 ) , it is assumed that the tendency will go on for all extrapolated values. If the population does non in fact follow the tendency after 2010, the theoretical account will non be valid at 2050 and will project/predict an wrong population.
Besides, many demographic factors have non been considered such as the population fluctuation due to future economic, environmental, and societal alteration. These factors can so be divided into the undermentioned variables that affect population growing: age construction of the population ( no. of adult females of kid bearing age ) , entire birthrate rate, wellness attention, instruction, occupations, life criterions, inmigration, outmigration ( alterations) , development, urbanization, disease, war/political turbulence and climate/natural catastrophes. Although there are may be issues with this theoretical account, it is based on the populations of 19502010. Since this period of clip did non hold any major events that would earnestly impact the population growing and the fact that the information is reasonably recent, it is accurate plenty to come to the decision that a population of 9 billion people by 2050 is merely when the rate of growing from 19502010 is taken into history.
Part B: Logarithmic Scales
Metric Prefix/Scientific Notation Conversions
Question 5 ( 2 KAPS )
Prefix 
Factor of each Prefix in Joules 
Energy ( E ) 
Energy ( E ) in Joules ( Scientific Notation ) 
Richter Magnitude ( R ) 
kilo ( KJ ) 
10^{3} 
130KJ 
Joule 
0.2 
mega ( MJ ) 
10^{6} 
4.9MJ 
Joule 
1.2 
giga ( GJ ) 
10^{9} 
2GJ 
Joule 
3.0 
11GJ 
Joule 
3.5 

63GJ 
Joule 
4.0 

tera ( TJ ) 
10^{12} 
16TJ 
Joule 
5.6 
63TJ 
Joule 
6.0 

180TJ 
Joule 
6.3 

360TJ 
Joule 
6.5 

peta ( PJ ) 
10^{15} 
2PJ 
Joule 
7.0 
11PJ 
Joule 
7.5 

63PJ 
Joule 
8.0 

210PJ 
Joule 
8.35 

840PJ 
Joule 
8.75 

exa ( EJ ) 
10^{18} 
2EJ 
Joule 
9.0 
11EJ 
Joule 
9.5 
Table 2 Metric Prefix/Scientific Notation Conversions
Graph 6 R vs. E
Graph 7 logR vs. Tocopherol
Table of Graphed Values
Richter Magnitude ( R ) 
Logarithm Base 10 of Energy () ( 5dp ) 
0.2 
5.11394J 
1.2 
Joule 
3.0 
Joule 
3.5 
Joule 
4.0 
Joule 
5.6 
Joule 
6.0 
Joule 
6.3 
Joule 
6.5 
Joule 
7.0 
Joule 
7.5 
16.04139J 
8.0 
=16.79934J 
8.35 
Joule 
8.75 
Joule 
9.0 
Joule 
9.5 
Joule 
Table 3 Table of Graphed Values
Graph 8 R vs. loge
It is apparent that the relationship between the Richter magnitude () and energy when a logarithm of base 10 is applied () is additive.
Graph 9 logR vs. loge
 To compose the relationship between R and E ( manually ) , the unknown constituents of the general logarithmic equation must be found utilizing the tabular array of values:
To compose the additive relationship betweenand( manually ) the general additive equation ( with modified variables ) and gradient must be found from the tabular array of values below:
…
Gradient Formula:
( For coordinates () ,)
VTable of Valuess 

Logarithm Base 10 of Energy () ( 5dp ) 
Richter Magnitude ( R ) 
5.11394J 
0.2 
6.69020J 
1.2 
9.30103J 
3.0 
10.04139J 
3.5 
10.79934J 
4.0 
13.20412J 
5.6 
13.79934J 
6.0 
14.25527J 
6.3 
14.55630J 
6.5 
15.30103J 
7.0 
16.04139J 
7.5 
16.79934J 
8.0 
17.32222J 
8.35 
17.92428J 
8.75 
18.30103J 
9.0 
19.04139J 
9.5 
Table 4 R V loge Table of Values
The undermentioned computation assumes that R is theaxis and thatis theaxis to guarantee that the relationship is every bit accurate as possible. Once the relationship is found, it is so rearranged to do R the topic of the equation.
Substituting ( 0.2, 5.11394 ) & A ; ( 9.0, 19.04139 ) into
( 6dp )
Since the gradient of the line has been found and a point may be taken from the tabular array of values above, the pointgradient expression is to be used:
( modified pointgradient signifier )
…
Substituting& A ;,
( Expanding )
( Simplifying )
From the computation above, the additive relationship can be modelled utilizing.
Rearranging to do R the topic of the equation ( for convenience when replacing energy values and so that it is in ‘’ signifier ) ,
()
[ Check ]
Substituting18.30103J
From Table 4,
( 6dp )
Although the substituted value did non give the exact Richter graduated table value due to, it is highly accurate. Therefore, the additive relationship between the Richter graduated table figure and the energy released () is justified.
Question 6 ( 1, 3,1.5 KAPS )
 To happen the Richter magnitude of the temblor that released 32PJ (J ) of energy,J needs to be substituted into the theoretical account found in inquiry 5 ( B ) ,
SubstitutingJoule,
()
()
( 2dp )
Therefore, the temblor that occurred in the Philippines and released 32PJ of energy had a Richter magnitude of about 7.8. The reasonability of this value can be checked utilizing portion of the Metric Prefix/Scientific Notation Conversions tabular array created in inquiry 5:
Prefix 
Energy ( E ) 
Richter Magnitude ( R ) 
peta ( PJ ) 
2PJ 
7.0 
11PJ 
7.5 

63PJ 
8.0 

210PJ 
8.35 

840PJ 
8.75 
Table 5 Reasonability Check Q6 ( a )
Since 32PJ is between 11PJ and 63PJ, the Richter magnitude of 32PJ must be between 7.5 and 8.0. As the Richter magnitude is 7.8, which is between 7.5 and 8.0, it is a sensible value for 32PJ of energy.
 To happen the energy release tantamount to a Richter magnitude of 5.5,demands to be substituted into the theoretical account found in inquiry 5 ( B ) ,
Substituting
( 4dp )
Sinceis tantamount to,
J ( 2dp )
Therefore the tantamount energy release to a Richter magnitude of 5.5 isJor 11.447878TJ ( 7dp ) .
From the tabular array 4 ( R vs. logE Table of Values ) , it can be seen that a Richter magnitude of 5.5 should let go of an energy degree under 16TJ ( or 16,000,000,000,000J ) and over 63,000,000,000J. Since 11,447,878,197,659.7J is in fact between this scope, it is a sensible value for the release of energy.
 To demo that an addition of 1 in Richter graduated table magnitude corresponds to about 30 times as much energy released, the energy release of two Richter scale magnitude with a difference of precisely 1.0 demand to be compared:
Prefix 
Factor of each Prefix in Joules 
Energy ( E ) 
Energy ( E ) in Joules ( Scientific Notation ) 
Richter Magnitude ( R ) 
giga ( GJ ) 
10^{9} 
2GJ 
Joule 
3.0 
11GJ 
Joule 
3.5 

63GJ 
Joule 
4.0 

tera ( TJ ) 
10^{12} 
16TJ 
Joule 
5.6 
63TJ 
Joule 
6.0 

180TJ 
Joule 
6.3 

360TJ 
Joule 
6.5 
Table 6 Q6 ( degree Celsius ) Energy/Richter Magnitude Table
( addition of 1.0 from 3.0 to 4.0 )
To happen how many times more energy is released from an event that measures 4.0 compared to 3.0 on the Richter graduated table, the energy released from a 4.0 event must be divided by the energy released from a 3.0 event ( if the order of magnitude is indistinguishable ) .
Richter Magnitude ( R ) 
Energy ( E ) 
0.2 
130KJ 
1.2 
4.9MJ 
3.0 
2GJ 
3.5 
11GJ 
4.0 
63GJ 
Table 7 Q6 ( degree Celsius ) Continued
Therefore, the energy released in a 4.0 Richter magnitude event is 31.5 times every bit big as the energy released in a 3.0 Richter magnitude event.
From this, it is apparent that an addition of 1 in the Richter graduated table does in fact correspond to an event with approximately 30 times every bit much energy released.
Sound Source 
Intensity () 
Intensity Level ( dubnium ) 
Jet plane from 30m 
100 
140 
Pain threshold 
3 
125 
Loud auto horn 
103 
90 
Door banging 
104 
80 
Normal conversation 
106 
60 
Quiet wireless 
108 
40 
Rustle of foliages 
1011 
10 
Threshold of hearing 
1012 
0 
Table 8 Sound Intensity and Intensity Level
Question 7 ( MAPS )
 Decibels are logarithmic units, and hence can non be added linearly like other figures. The expression below shows the relationship between sound strength ( I ) and intensity degree ( L ) .
Where L is the strength degree measured in dubnium,is the ( entire ) sound strength measured in Watts per square meter () andis the threshold of hearing () , L is defined as the followers:
… ( Maths Quest Maths B Year 11 for QLD )
To happen the strength degree ( L ) of a normal conversation happening while a quiet wireless is playing, the entire sound strength ofmust be substituted into,
Substituting,
60.04dB ( 2dp )
Therefore, the sound degree of a normal conversation and a quiet wireless playing is tantamount to 60.04dB. However, if these sound degrees were to be added linearly, the consequence will be wrong. A important drawback of the dB system is the inability of dBs to be added linearly which makes it more hard to happen the entire sound degree of many beginnings of sound happening at the same time without anterior cognition of logarithmic graduated tables.
 Since the scope of sound of a concert is comparatively little compared to the threshold of human hearing, the graduated table used for the concert can be truncated to guarantee maximal truth.
To happen an alternate sound graduated table with a scope of 1 to 10,
The general signifier of the new graduated table ( N ) based on the sound strength ( I )
LetTungstenandon the new graduated table,
( 5dp )
Substituting
( –)
( 5dp )
Substitutingandinto,
Therefore the mathematical theoretical account for the new graduated table to be used in a local concert hall is( where S is the sound strength andis the matching value on the new graduated table ) .
Strengths 
Failings 
This theoretical account is simple and easy to utilize. 
It does non give the performing artists an thought of the existent sound strength of their concert. 
The sound of the concert can be communicated rapidly to the performing artists, which means that any alterations in volume can be implemented every bit shortly as possible. 
Since normal conversation is 4.44 ( 2dp ) on the new graduated table, it is likely that most of the manus marks will be concentrated in the scope of 510. This could take to error and trouble when pass oning the volume of the concert to the performing artists 
The graduated table is more suited for the state of affairs than dBs. 
Valuess on the new graduated table can non be summed linearly. 
Table 9 New Scale: Strengths and Failings
From inquiries 5 and 7, it can be seen that are many advantages and some disadvantages in utilizing the Richter graduated table and the Decibel graduated table instead than utilizing magnitude/intensity degrees. The tabular array below provides a sumup of strengths and restriction of these logarithmic graduated tables:
Logarithmic Scales Commonly Used – Richter Scale and Decibel System 

Strengths 
Restrictions 
Provides a graduated table with values that most people can grok. 
Does non give an indicant of the magnitude/intensity of the event for people without a background apprehension of logarithmic graduated tables. 
Supply a additive graduated table which impacts truth 
Both Richter graduated table values and dB values can non be added linearly which may ensue in incommodiousness when trying to sum values on these graduated tables ( this largely refers to ciphering the entire dB value of two sound beginnings happening at the same time ) 
Brands it clear that the rate of alteration is changeless ( as seen in the additive Graph 8 R vs. loge in 5a ) 

Corrects skewness towards big values and hence makes it easier to utilize and read graphs 

Table 10 Strengths and Limitations of the Richter Scale and the Decibel System
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