Add Maths Project Work (2013)

7/28/2019 Add Maths Project Work (2013)

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ADDITIONAL

MATHEMATICSPROJECT WORK 1/2013

SEKOLAH MENENGAH KEBANGSAANPUCHONG BATU (14),SELANGOR D.E

TITLE: LOGARITHMS

NAME:SASIVARNEN A/L GUNASEKARAN

FORM:5 AMANAH

I/C NUMBER: 961124-14-5619

TEACHER:MISS.CHEN


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OBJECTIVESThe aims of carrying out this projects works are:

i) To apply and adapt a variety of problem solving strategies to solve problems

ii) To improve thinking skillsiii) To promote effective mathematical communicationiv) To develop mathemathical knowledge through problem solving in a way

that increases students interests and confidencev) To use language of mathemathics to express mathemathical idea

precisely

vi)

To provide learning environment that stimulates and enhance effectivelearningvii) To develop positive attitude toward mathemathics


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ACKNOWLEDGEMENT

I would like to express my special thanks of gratitude to my teacher Miss.Chen Nyuk as well as our principal Pn.Zuhairah Binti Yusuf whogave me the golden opportunity to do this wonderful project on the topiclogarithm,which also helped me in doing a lot of researchs and I came toknow about so many new things where I am really thankful to them.Iappreciate their patience in guiding me to complete this project work.

Secondly, I would also like to thank my mother,Punithavathi a/pRamasamy specifically for guiding me throughout this project.She gaveme full support in this project work both financially and mentally.I takethis opportunity to express my profound gratitude and deep regards tomy mother for her exemplary guidance, monitoring and constantencouragement throughout the course of this project.The blessing, helpand guidance given by her time to time shall carry me a long way in the

journey of life on which I am about to embark.

I also take this opportunity to express a deep sense of gratitude to my

fellow friends for their cordial support, valuable information andguidance, which helped me in completing this task through variousstages.I am obliged to them for the valuable information provided bythem in their respective fields. I am grateful for their cooperation duringthe period of my assignment.

Last but not least,I would like to thank everyone who came in handy for helping me completing this project work.Again,thank you very much


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INTRODUCTION

One of the mathematical concepts which we must be familiar with islogarithm.Before the days of scientific calculators,logarithms were use used tomultiply or divide extreme numbers using mathematical tables.For thecalculations,ten was the most common base to use.Logarithm to the base of ten is also called the common logarithm.Other base such as two,five,eight canalso be used.The ancient Babylonians had used bases up to 60.

Logarithms have many applications in various fields of studies.In the early17th century,it was rapidly adopted by navigators,scientists,engineers andastronomers to perform computations more easily using slide rules andlogarithm tables.


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History on LogarithmWhat is a logarithm? Ask a modern mathematician nowadays and you will get a verydifferent answer from the one you might have got from a mathematician several centuriesago. Indeed, even the very first mathematicians who worked with the logarithmic relationwould have given an explanation that would seem quite foreign to a modernmathematician. So how did the logarithmic relation come about, and how is it that theconcept underwent so much change? We will address these questions by looking at theemergence of this concept, and examining some of the issues surrounding its origins.

In fact, the question of the origins of the logarithmic relation does not have a simpleanswer. At least two scholars, the Scottish baron John Napier (1550-1617) and Swisscraftsman Joost Brgi (1552-1632), produced independently systems that embodied thelogarithmic relation and, within years of one another, produced tables for its use. Thisparallel insight is fascinating and rich in historical detail, and it reveals somemethodological challenges for historians of mathematics. In light of all this, we will examine

the ideas of these two scholars, as well as explore how historians have portrayed thisintricate situation and the questions it raises about mathematics.

In large part, we intend to re-introduce teachers to a concept that is often taught without any reference to its original appearance on the mathematical scene. We hope that a closeexamination of Napier's and Brgi's conceptions will enable teachers to consideralternative placement for introducing the idea of logarithms as part of or after a unit onsequences. Furthermore, we provide in what follows mathematical and historical content,as well as student exercises, to promote the teaching of the logarithmic relation from itshistorical roots, which are firmly situated in simultaneous consideration of arithmetic andgeometric sequences.The logarithmic relation, captured in modern symbolic notation as

log( a b)=log( a )+log( b),

is useful primarily because of its power to reduce multiplication and division to the lessinvolved operations of addition and subtraction. When this relation hit the scene in theearly seventeenth century, its impact was substantial and immediate. Modern historians of mathematics, John Fauvel and Jan van Maanen (2000) , illustrate this vividly:

When the English mathematician Henry Briggs learned in 1616 of the invention of logarithms by John Napier, he determined to travel the four hundred miles north toEdinburgh to meet the discoverer and talk to him in person.

Indeed, Fauvel and van Maanen assert that the meeting of Briggs and Napier is one of thegreat tales in the history of mathematics.Unfortunately, it seems that many teachers (andtheir students) are not aware of this particular great tale or, at most, superficiallyassociate the names Briggs and Napier with the invention of the logarithm. Typically, thesegroups know little about the original conceptions of the logarithmic relation.

An anecdote concerning a conversation between Fauvel (1995) and a colleague reinforcesthis. Fauvel recounted that when he inquired of his colleague how to teach logarithms, thecolleague responded, Whatever for? Surely no one needs to learn about those any more,
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now that we have calculators and com puters (p. 39). Many teachers, when approachedabout the possibility of teaching logarithms using a historical context, may express thesame opinion. Fauvel's counter- argument to his colleague was that logarithms are a goodand accessible example of something fundamentally changing its conceptual role withinmathematics . Indeed, examining the historical development of the logarithm with

students by exploring arithmetic and geometric progressions allows students a moredeeply rooted understanding of wha t is going on

The modern concept of the logarithm typically appears late in a second algebra orprecalculus course (grades 10 or 11 in the US), situated after a study of polynomial andrational functions, but before sequences and series and conic sections. This organizationoften leaves students with an impression of disconnectedness between mathematicaltopics. Complicating matters further, the logarithm is often presented only briefly in suchan algebra or precalculus course, in order to lead to a broader study of logarithmicfunctions to match students' previous study of other functions (e.g., linear, polynomial,rational). Lastly, in order to focus on the study of logarithmic functions, instruction andcurricular organization dictate that this function exist as the inverse of the exponentialfunction. This, in particular, contrasts starkly with the historical circumstances: in fact, thetrajectory of modern mathematical pedagogy does not imitate history, as exponentialsarrived on the mathematical scene well after the introduction of the logarithm!

Victor Katz (1995; 1997) provided a succinct argument for examining the development of logarithms from a historical perspective. He observed that Napier developed logarithmsfor use in the extensive plane and spherical trigonometrical calculations necessary forastronomy.Although the motivation for developing logarithms is significant, Katz notedthat, in general, students today often know very little about astronomy and about themagnitude of both the numbers and the calculations involving such numbers that werenecessary to advance the science of astronomy. Astronomical advances have remainedcritical throughout civilization, however, and Katz (1997 ) indicated that, it is well for us tointroduce it [astronomy] whenever possible.

Although there is a strong temptation simply to present the definition and severalproperties of the logarithm and exercises to practice each, we propose that incorporatingoriginal and parallel insights of the logarithm can enrich instruction and learning of thetopic, both for this concept and more broadly for a student's understanding of mathematicsand its relations and development.

The late sixteenth century saw unprecedented development in many scientific fields;notably, observational astronomy, long-distance navigation, and geodesy science, or effortsto measure and represent the earth. These endeavors required much from mathematics.For the most part, their foundation was trigonometry, and trigonometric tables, identities,and related calculation were the subject of intensive enterprise. Typically, trigonometricfunctions were based on non-unity radii, such as R=10,000,000, to ensure precise integeroutput.Reducing the calculation burden that resulted from dealing with such large numbersfor practitioners in these applied disciplines, and with it, the errors that inevitably crept into the results, became a prime objective for mathematicians. As a result, much energy andscholarly effort were directed towards the art of computation.
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Accordingly, techniques that could bypass lengthy processes, such as long multiplicationsor divisions, were explored. Of particular interest were those that replaced theseprocesses with equivalent additions and subtractions. One method originating in the latesixteenth century that was used extensively to save computation was the techniquecalled prosthaphaeresis, a compound constructed from the Greek

terms prosthesis (addition) and aphaeresis (subtraction). This relation transformed longmultiplications and divisions into additions and subtractions via trigonometric identities,such as:

2cos( A)cos( B)=cos( A+B)+cos( AB).

When one needed the product of two numbers x and y , for example, trigonometric tableswould be consulted to find A and Bsuch that:

x =cos( A)and y =cos( B).With A and B determined, cos( A+B) and cos( AB) could be read from the table and half of the sum taken to find the original product in question. Thus the long multiplication of twonumbers could be replaced by table look-up, addition, and halving. Such rules wererecognized as early as the beginning of the sixteenth century by Johannes Werne r in 1510,but their application specifically for multiplication first appeared in print in 1588 in a work by Nicolai Reymers Ursus (Thoren, 1988) .Christopher Clavius extended the methodsof prosthaphaeresis, of which examples can be found in his 1593 Astrolabium (Smith, 1959 0

Finally, with the scientific community focused on developing more powerful computationalmethods, the desire to capture symbolically essential mathematical ideas behind thesedevelopments was also growing. In the fifteenth and sixteenth centuries, mathematicianssuch as Nicolas Chuquet (c. 1430 1487) and Michael Stifel (c. 1487 1567) turned theirattention to the relationship between arithmetic and geometric sequences while workingto construct notation to express an exponential relationship. The focus on mathematical

symbolism in centuries prior and the growing attention to notation particularly theexperimentation with different versions of exponent notation played a critical role in therecognition and clarification of such a relationship. Now the mathematical connectionbetween a geometric and an arithmetic sequence could be made all the more apparent bysymbolically capturing these sequences as successive exponential powers of a givennumber and the exponents themselves, respectively (see Figure 6) . The work on therelationships between sequences was mathematically important per se, but was equallysignificant for providing the inspiration for the development of the logarithmic relation.

John Napier Introduces LogarithmsIn such conditions, it is hardly surprising that many mathematicians were acutely aware of the issues of computation and were dedicated to relieving practitioners of the calculationburden. In particular, the Scottish mathematician John Napier was famous for his devices toassist with computation. He invented a well-known mathematical artifact, the ingeniousnumbering rods more quaintly known as Napier's bones, that offered mechanical meansfor facilitating computation. (For additional information on Napier's bones, see thearticle, John Napier: His Life, His Logs, and His Bones (2006) .) In addition, Napierrecognized the potential of the recent developments in mathematics, particularly thoseof prosthaphaeresis, decimal fractions, and symbolic index arithmetic, to tackle the issue of
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reducing computation. He appreciated that, for the most part, practitioners who hadlaborious computations generally did them in the context of trigonometry. Therefore, aswell as developing the logarithmic relation, Napier set it in a trigonometric context so it would be even more relevant.

Napier first published his work on logarithms in 1614 under the title Mirifici logarithmorum canonisdescriptio, which translates literally as A Description of the Wonderful Table of Logarithms. Indeed, thevery title Napier selected reveals his high ambitions for this technique---the provision of tables basedon a relation that would be nothing short of wonder - working for practitioners. As well as providing ashort overview of the mathematical details, Napier gave technical expression to his concept. He coineda term from the two ancient Greek terms logos, meaning proportion, and arithmos, meaning number;compounding them to produce the word logarithm. Napier used this word as well as the designations

natural and artificial for numbers and their logarithms, respectively, in his text.

Despite the obvious connection with the existing techniques of prosthaphaeresis and sequences,Napier grounded his conception of the logarithm in a kinematic framework. The motivation behind thisapproach is still not well understood by historians of mathematics. Napier imagined two particles

traveling along two parallel lines. The first line was of infinite length and the second of a fixed length(see Figures 2 and 3). Napier imagined the two particles to start from the same (horizontal) positionat the same time with the same velocity. The first particle he set in uniform motion on the line of infinite length so that it covered equal distances in equal times. The second particle he set in motionon the finite line segment so that its velocity was proportional to the distance remaining from theparticle to the fixed terminal point of the line segment.


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Figure 2. Napier's two parallel lines with moving particles

Napier generated numerical entries for a table embodying this relationship. He arranged his table bytaking increments of arc minute by minute, then listing the sine of each minute of arc, and then itscorresponding logarithm. However in terms of the way he actually computed these entries, he wouldhave in fact worked in the opposite manner, generating the logarithms first and then choosing those

that corresponded to a sine of an arc, which accordingly formed the argument. For example, he wouldhave computed values that appear in the first column of Table 1 via the relation:

pn+1= pn(1110 7)where p0=10 7.

pn n=log nap ( pn) Corresponding angle ( )

10000000.0000000 0 90 00

9999999.0000000 1 89 59

9999998.0000001 2 89 58

9999997.0000003 3

9999996.0000006 4 89 57

9999995.0000010 5

9999994.0000015 6

9999993.0000021 7 89 56

9999992.0000028 8

9999991.0000036 9

9999990.0000045 10

9999989.0000055 11 89 55

Table 1. Napier's logarithms

The values in the first column (in bold) that corresponded to the Sines of the minutes of arcs (thirdcolumn) were extracted, along with their accompanying logarithms (column 2) and arranged in thetable. The appropriate values from Table 1 can be seen in rows one to six of the last three columns inFigure 4. Napier tabulated his logarithms from 0 to 45 in minutes of arc, and by symmetry providedvalues for the entire first quadrant. The excerpt in Figure 4 gives the first half of the first degree and,by symmetry, on the right the last half of the eighty-ninth degree.


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To complete the tables, Napier computed almost ten million entries from which he selected theappropriate values. Napier himself reckoned that computing this many entries had taken him twentyyears, which would put the beginning of his endeavors as far back as 1594.

Figure 4. The first page of Napier's tables(Image used courtesy of Landmarks of Science Series, NewsBank-Readex)

Napier frequently demonstrated the benefits of his method. For example, he worked through aproblem involving the computation of mean proportionals, sometimes known as the geometric mean.He reviewed the usual way in which this would have been computed, and pointed out that histechnique using logarithms not only finds the answer earlier (that is, faster!), but also uses only oneaddition and one division by two! He stated:

At about the same time in Switzerland, Joost Brgi, a court clock maker by profession, grappled withthe same issues of computation. Brgi's key motivation was not only to facilitate computation, butalso to produce a single table that could be applied to all arithmetical operations, rather than needingvarious tables to perform them all. In his work, Arithmetische und Geometrische ProgressTabulen ( Arithmetic and Geometric Progression Tables), published in 1620, Brgi noted that havingseparate tables for multiplication, division, square roots, and cube roots is not alone irksome, butalso laborious and cumbersome (Preface, 1, xi -xii).

Furthermore, Brgi grounded his conception directly in the relation between two progressions. Hestated that he was able to create one table for a multiplicity of calculations by considering two self -producing and corresponding progressions (Preface, 1, xv -xvi): one arithmetic and the othergeometric. To illustrate the underlying principle by means of nice numbers, he gave correspondingprogressions based on the powers of two, as shown in Figure 6.

Brgi gave the relation of powers of two as an example, but in fact different parameters underpinnedhis logarithmic relation. As he noted, successive powers of two increase too quickly to be useful tointerpolate between values so instead he used a common ratio of 1.0001 , and the successive valueswere tabulated as follows:

bn+1=bn(1.0001)where b0=10 8
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Thus, each successive value in the table can be generated by multiplying the previous one by 1.0001 .Brgi used the factor of 108 to allow for greater integer precision.For example, the logarithm of 101907877 (a black number) can be found using the tables as follows(we must note that his logarithm values increased by 10 and were also multiplied by a factor of 10 ).

Logarithms represented at this time in so many ways both what was old and what was new. This

relation looked back to reflect concerns of computation, but looked forward to nascent notions aboutmathematical functions. Although logarithms were primarily a tool for facilitating computation, theywere but another of the crucial insights that directed the attention of mathematical scholars towardsmore abstract organizing notions. But one thing is very clear: the concept of logarithm as weunderstand it today as a function is quite different in many respects from how it was originallyconceived. But eventually, through the work, consideration, and development of manymathematicians, the logarithm became far more than a useful way to compute with large unwieldynumbers. It became a mathematical relation and function in its own right.

In time, the logarithm evolved from a labor saving device to become one of the core functions inmathematics. Today, it has been extended to negative and complex numbers and it is vital in manymodern branches of mathematics. It has an important role in group theory and is key to calculus, with

its straightforward derivatives and its appearance in the solutions to various integrals. Logarithmsform the basis of the Richter scale and the measure of pH, and they characterize the music intervals inthe octave, to name but a few applications. Ironically, the logarithm still serves as a labor savingdevice of sorts, but not for the benefit of human effort! It is often used by computers to approximatecertain operations that would be too costly, in terms of computer power, to evaluate directly,particularly those of the form xn.


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Application of logarithms in two different field of study.

Application #1::pH Measurement

pH is a measure of acidity or alkanity and is surprisingly common measurement.For example,in the chemical industry,the acidity of the reagents in many types of reactor has to becontrolled to enable optimum reaction conditions.In addition,in the water industry,the acidity of water for consumption and of effluent for discharge have to be controlled carefully to satisfylegislative requirements.

pH is an electro-chemical measurement,invariably made by means of the so called glasselectrode.It is notoriously difficult measurement to make because of factors such as drift andfouling.Understanding the significance of measurements requires an appreciation of electro-chemical equilibria.Using pH for control puposes is problematic because of the inherent non-linearities and time delays.

The formal definition of pH is given by:

pH= -log 10[H+]

where [ ] denotes concentration of ions in aqueous solutions with units of g ions/L.In the case of hydrogen,whose atomic and ionic weights are same,[H +] has units of g/L or kg m -3.Thelogarithmic scale means that pH increase one unit for each decrease by a factor of 10 in [H +].

Pure water dissociates very weakly to produce hydrogen and hydroxyl ions according to

H2O H+ + OH -

At equilibrium at approximately 23 oC,their concentration are such that

[H +] + [OH -] =10 -14

The diassociation must produce equal concentrations of H + and OH - ions,so

[H +] = [OH -] =10 -7

Since pure water is neutral,by definition,it follows that for neutrality:

pH water = -log 10[10 -7]=7

This gives rise to familiar pH of 0-14,symmetrical about pH 7,of which 0-7 corresponds to acidicsolutions and 7-14 to alkaline solutions.To evaluate pH of alkaline solutions,it is usual to

substitute for H+

in the above equations: pH alkali = -log10 10 -14 = 14 + log 10[OH -]

[OH -]

Example: a) Suppose that you test apple juice and find that the hydrogen ionconcentration is[H +] = 0.0003. Find the pH value and determine whether the juice is basicor acidic.


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b) You test some ammonia and determine the hydrogen ion concentration to be [H +] = 1.3 10 9. Find the pH value and determine whether the ammonia is basic or acidic.In eachcase, I need to evaluate the pH function at the given value of [H +].

a) In the case of the apple juice, the hydrogen ion concentration is [H +] = 0.0003, so:

pH = log [H +] = log [0.0003] = 3.52287874528.....which is less than 7, so this isacidic.

b) In the case of the ammonia, the hydrogen ion concentration is [H +] = 1.3 10 9, so:

pH = log [H +] = log [1.3 10 9] = 8.88605664769......which is more than 7, sothis is basic.

EXAMPLE ::Neutralization Control

Neutralization is the process whereby acid and base reagents are mixed to produce a productof specified pH.In the context of waste water and effluent treatement the objective is to adjust the

pH to a value of 7 altough,in practice,any value in the range 6-8 is good enough.In manychemical reactions the pH has to be controlled at a value other than 7,which could be anywherein the range of 0-14.Neutralization is always carried out in aqueous solutions,pH is ameaningless quantity otherwise.Note that a base that is soluble in water is usually referred to asan alkali.

pH is without doubt the most difficult or common process variable to control.For example,themeasurement is electrochemical,made with a glass and reference electrode pair as show

below,and is prone to contamination,hysteresis and drift.The signal produced,beinglogarithmic,is highly non linear.The process being controlled invariably has a wide range of

both concentrations and flow rate.The rangeability of flow gives rise to variable residencetimes.To achieve satisfactory control,all of the issues have to be addresses.

A pH sensing instrument


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Application #2::Radioactivity

In his article Light Attenuation and Exponential Laws in the last issue of Plus, Ian Garbett discussed the phenomenon of light attenuation, one of the many physicalphenomena in which the exponential function crops up. In this second article hedescribes the phenomenon of radioactive decay, which also obeys an exponential law,and explains how this information allows us to carbon-date artefacts such as the DeadSea Scrolls.

In the previous article, we saw that light attenuation obeys an exponential law. To showthis, we needed to make one critical assumption: that for a thin enough slice of matter,the proportion of light getting through the slice was proportional to the thickness of theslice.

Exactly the same treatment can be applied to radioactive decay. However, now the"thin slice" is an interval of time, and the dependent variable is the number of radioactive atoms present, N ( t ).

Radioactive atoms decay randomly. If we have a sample of atoms, and we consider atime interval short enough that the population of atoms hasn't changed significantlythrough decay, then the proportion of atoms decaying in our short time interval will beproportional to the length of the interval. We end up with a solution known as the "Lawof Radioactive Decay", which mathematically is merely the same solution that we sawin the case of light attenuation. We get an expression for the number of atomsremaining, N , as a proportion of the number of atoms N 0 at time 0, in terms of time, t :

N / N 0 = e- lt ,

where the quantity l , known as the "radioactive decay constant", depends on theparticular radioactive substance.

Again, we find a "chance" process being described by an exponential decay law. We caneasily find an expression for the chance that a radioactive atom will "survive" (bean original element atom) to at least a time t . The steps are the same as in the case of photon survival.
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Mean lifetime of a Radioactive AtomOn average, how much time will pass before a radioactive atom decays?

This question can be answered using a little bit of calculus. Suppose that we invert ourfunction for N / N 0 in terms of t , to get an expression for t as a function of N / N 0 . Once wehave an expression for t , a "definite integral" will give us the mean value of t (this ishow "mean value" is defined).

From the equation above, taking logarithms of both sides we see that lt = -ln( N / N 0 ) =ln( N 0 / N ), so our equation for t is

t plotted against F

For convenience, we'll now write F for N / N 0 . Note that that the domain of F is theinterval from zero to 1, which corresponds to the interval of time from zero to infinity.Plotting t against F with a value of l =1 gives the graph on the right.

To find < t >, the mean value of time of survival, all we have to do is find the integral


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which is a very tidy result!

Incidentally, our formula for t gives us an easy way of finding the half-life , the time ittakes for half the nuclei in a sample to decay. The half-life (often denoted t 1/2 ) is

just t (1/2) = (1/ l ) ln(2). The equivalent thickness for the medium in radiationattenuation is known as "half-value thickness". Similarly, in a population which growsexponentially with time there is the concept of "doubling time".

Libby's LegacyWe started the first article by talking about carbon dating and the Dead Sea scrolls.Let's look further at the technique behind the work that led to Libby being awarded aNobel prize in 1960.

Carbon 14 (C-14) is a radioactive element that is found naturally, and a living organismwill absorb C-14 and maintain a certain level of it in the body. This is because there iscarbon dioxide (CO 2 ) exchange in the atmosphere, which leads to constant turnover of carbon molecules within the body cells.

Once an organism dies there is no further CO 2 exchange, and so the ratio of C-14 to thefar more common carbon isotope, C-12, will begin to decrease as the C-14 atomsdecay, yielding nitrogen (N-14) with the emission of an electron (or "beta particle") plusan anti-neutrino.

The ratio of C-14 to C-12 in the atmosphere's carbon dioxide molecules is about1.310 -12 , and this value is assumed constant for the main part of archaeologicalhistory since the formation of the earth's atmosphere.

Knowing the level of activity of a sample of organic material enables us to deduce howmuch C-14 there is in the material at present. Since we also know the ratio of C-14 toC-12 originally, we can find the time that has passed since carbon exchange ceased,that is, since the organic material "died".

In the case of the Dead Sea scrolls, important questions required answers. Were theyforgeries? Did they really date from around the time of Christ? Before or after?

Using Libby's radiocarbon dating technique, the scrolls have been dated, using the linencoverings the scrolls were wrapped in. One scroll, the Book of Isaiah, has been dated at1917BC 275 years, certainly long before the time of Christ. Some of the others areroughly contemporary with Christ.

Let's take a look at an example of how dates are calculated using Libby's method.


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Suppose a linen sample of 1 gram is analysed in a counter. The activity is measured atapproximately 11.9 decays per minute. We'll denote the magnitude of the rate of decayof the Carbon 14 nuclei as R. This magnitude is equal to the rate that beta particles aredetected. So

Recall that the exponential law for the number of Carbon 14 nuclei present says that

and so

which tells us that R= lN , and that the activity at t =0 (the time the linen wasmanufactured) is R(0) = lN 0 .

Substituting gives us an exponential relation in terms of the measured activity:

Now the decay constant for Carbon-14 is l = 3.8394 10 -12 per second. Thiscorresponds to a half life of 5,730 years.

We can calculate the number of Carbon-12 nuclei in 1 gram of carbon:

Using the (living) ratio of C-14 to C-12, this implies that the original ( t = 0) number of Carbon 14 nuclei was

Now, rearranging the exponential activity law gives


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R0 is simply (3.839410-12 )(6.522110 10 ) = 0.2504 decays per second. The measured

rate is R( t ) = 11.9 decays per minute = 0.1983 decays per second.

The value for t that results is

which is approximately 1929 years. This an approximate age for some of the scrolls.

In a similar way, dating charcoal found at Stonehenge gives ages of approximately

3798 275 years, and, when used on some of the oldest archaeological artefacts in theAmericas, the technique gives ages of approximately 25,000 years, corresponding to atime of significantly lower sea levels and supporting the theory that the very firsthumans in the "New World" crossed the Bering Straits by foot from Siberia into Alaska.

Incidentally, other "larger time scale" radio-dating techniques exist, apart from Libby'sradiocarbon method. Here isotopes with longer half lives are used, which enables datingof geological formations and rocks. However, the essential ideas are analogous. Forexample, in lava form, molten lead and Uranium-238 (standard isotope) are constantlymixed in a certain ratio of their natural abundance. Once solidified, the lead is "locked"in place and since the uranium decays to lead, the lead-to-uranium ratio increases withtime. In this way, some of the oldest rocks have been measured at approximately 3billion years.

Radioactive decay rates

The decay rate , or activity , of a radioactive substance are characterized by:

Constant quantities :

The half-life t 1/2 , is the time taken for the activity of a given amount of aradioactive substance to decay to half of its initial value; see List of nuclides.

The mean lifetime , " tau " the average lifetime of a radioactive particle before

decay. The decay constant , "lambda " the inverse of the mean lifetime.

Although these are constants, they are associated with statistically random behavior of populations of atoms. In consequence predictions using these constants are lessaccurate for small number of atoms.
http://en.wikipedia.org/wiki/Half-lifehttp://en.wikipedia.org/wiki/Half-lifehttp://en.wikipedia.org/wiki/List_of_nuclideshttp://en.wikipedia.org/wiki/Mean_lifetimehttp://en.wikipedia.org/wiki/Mean_lifetimehttp://en.wikipedia.org/wiki/Tauhttp://en.wikipedia.org/wiki/Decay_constanthttp://en.wikipedia.org/wiki/Decay_constanthttp://en.wikipedia.org/wiki/Lambdahttp://en.wikipedia.org/wiki/Lambdahttp://en.wikipedia.org/wiki/Decay_constanthttp://en.wikipedia.org/wiki/Tauhttp://en.wikipedia.org/wiki/Mean_lifetimehttp://en.wikipedia.org/wiki/List_of_nuclideshttp://en.wikipedia.org/wiki/Half-life


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In principle the reciprocal of any number greater than one a half-life, a third-life, oreven a (1/ 2)-life can be used in exactly the same way as half-life; but the half-life t 1/2 is adopted as the standard time associated with exponential decay.

Time-variable quantities :

Total activity A , is number of decays per unit time of a radioactive sample. Number of particles N , is the total number of particles in the sample. Specific activity S A , number of decays per unit time per amount of substance of the

sample at time set to zero ( t = 0). "Amount of substance" can be the mass, volumeor moles of the initial sample.

These are related as follows:

where N 0 is the initial amount of active substance substance that has the samepercentage of unstable particles as when the substance was formed.

Units of radioactivity measurements

The SI unit of radioactive activity is the becquerel (Bq), in honor of the scientist HenriBecquerel. One Bq is defined as one transformation (or decay or disintegration) per

second. Since sensible sizes of radioactive material contains many atoms, a Bq is a tinymeasure of activity; amounts giving activities on the order of GBq (gigabecquerel, 1 x10 9 decays per second) or TBq (terabecquerel, 1 x 10 12 decays per second) arecommonly used.

Another unit of radioactivity is the curie, Ci, which was originally defined as the amountof radium emanation (radon-222) in equilibrium with one gram of pure radium, isotope Ra-226. At present it is equal, by definition, to the activity of anyradionuclide decaying with a disintegration rate of 3.7 10 10 Bq, so that 1 curie (Ci) =3.7 10 10 Bq. The use of Ci is currently discouraged by the SI. Low activities are alsomeasured in disintegrations per minute (dpm).

Universal law of radioactive decay

Radioactivity is one very frequent example of exponential decay. The law describes thestatistical behavior of a large number of nuclides, rather than individual ones. In thefollowing formalism, the number of nuclides or nuclide population N , is of course adiscrete variable (a natural number )but for any physical sample N is so large(amounts of L = 10 23, avagadro's constant ) that it can be treated as a continuous
http://en.wikipedia.org/wiki/Number_of_particleshttp://en.wikipedia.org/wiki/International_System_of_Unitshttp://en.wikipedia.org/wiki/Becquerelhttp://en.wikipedia.org/wiki/Henri_Becquerelhttp://en.wikipedia.org/wiki/Henri_Becquerelhttp://en.wikipedia.org/wiki/Curiehttp://en.wikipedia.org/wiki/Radiumhttp://en.wikipedia.org/wiki/Isotopehttp://en.wikipedia.org/wiki/Curiehttp://en.wikipedia.org/wiki/Exponential_decayhttp://en.wikipedia.org/wiki/Natural_numberhttp://en.wikipedia.org/wiki/Avagadro%27s_constanthttp://en.wikipedia.org/wiki/Avagadro%27s_constanthttp://en.wikipedia.org/wiki/Natural_numberhttp://en.wikipedia.org/wiki/Exponential_decayhttp://en.wikipedia.org/wiki/Curiehttp://en.wikipedia.org/wiki/Isotopehttp://en.wikipedia.org/wiki/Radiumhttp://en.wikipedia.org/wiki/Curiehttp://en.wikipedia.org/wiki/Henri_Becquerelhttp://en.wikipedia.org/wiki/Henri_Becquerelhttp://en.wikipedia.org/wiki/Becquerelhttp://en.wikipedia.org/wiki/International_System_of_Unitshttp://en.wikipedia.org/wiki/Number_of_particles


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variable. Differential calculus is needed to set up differential equations for modelling thebehaviour of the nuclear decay.

EXAMPLE::Nuclear Reactors

A device used to generate power, in which nuclear fission takes place as a controlledchain reaction, producing heat energy that is generally used to drive turbines andprovide electric power. Nuclear reactors are used as a source of power in large powergrids and in submarines.A Closer Look A nuclear reactor uses a nuclear fission chain reaction to produce energy.The cylindrical core of a reactor consists of fuel rods containing pellets of fissionablematerial, usually uranium 235 or plutonium 239. These unstable isotopes readily splitapart into smaller nuclei (in the fission reaction) when they absorb a neutron; theyrelease large quantities of energy upon splitting, along with more neutrons that may beabsorbed by the nuclei of other isotopes, causing a chain reaction. The neutrons areexpelled from the fission reaction at very high speeds, and are not likely to be absorbedat such speeds.

Moderators such as heavy water are therefore needed to slow the neutrons to a speedat which they are readily absorbed. The fuel rods contain enough fissionable materialarranged in close enough proximity to start a self-sustaining chain reaction. To regulatethe speed of the reaction, the fuel rods are interspersed with control rods made of amaterial (usually boron or cadmium) that absorbs some of the neutrons given off by thefuel. The deeper the control rods are inserted into the reactor core, the more thereaction is slowed down. If the control rods are fully inserted, the reaction stops. Thechain reaction releases enormous amounts of heat, which is transferred through aclosed loop of radioactive water to a separate, nonradioactive water system, creatingpressurized steam. The steam drives turbines to turn electrical generators.

Nuclear Reactor


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PART 2


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The volume,V,in cm 3,of a solid sphere and its diameter,D,in cm,are related by the equationV=mD n,where m and n are constants.You can find the value of m and n by conducting theactivities below.

A i) Choose six different spheres with diameters between 1 cm to 8 cm.Measure the diameters of the six spheres using a pair of vernier calipers.

ii) Find the volume of each sphere without using the formula of volume.(You can use the apparatus in the science lab to help you)

Archimedess principle states that the upward buoyant force is exerted on a body immersed in afluid is equal to the weight of the fluid the body displaces.In other words,an immersed object is

buoyed up by a force equal to the weight of the fluid it actually displaces

Using the method of water displacement and knowing the water density is 1g/cm 3,the volume of a solid sphere is equivalent to the weight of the water displaced (in grams),and thus it can bemeasured

iii) Tabulate the values of the diameter ,D,in cm,and its corresponding volume,V,in cm3.

Sphere Diameter (cm)

1 1.02 2.43 3.84 5.25 6.66 8.0

Sphere Volume(cm 3)

1 0.52362 7.23823 28.73104 73.62235 150.53296 268.0832


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B Find the value of m and nusing logarithms with any two sets of values obtained in the tableabove.

Given that the volume,V,of a solid sphere is

V=mD n

Taking logarithm on both sides yield which can be expanded to

log10V=log 10m + log 10Dn

log 10V=log 10m + n log 10D (2)

Using logarithms with any two sets of values s 1(D1V1) and s 2 (D1 V1),we have

log 10V1=log 10m + n log 10D1 (3) log 10V2=log 10m + n log 10D2 (4)

From Eq.(3) and (4),it is obvious tha log 10m can be eliminated by taking (3) -(4)log10V1- log 10V2= n log 10D1- n log 10D2

which can be simplified by the Laws of Logarithms

log10 V1 = n log 10 D1V2 D2

Dividing both sides with log 10 D1 , we can determine the value of nD2

Sphere Diameter (cm 3)

Volume(cm 3)

1 1.0 0.52362 2.4 7.23823 3.8 28.73104 5.2 73.62235 6.6 150.5329

6 8.0 268.0832

Sphere Diameter (cm 3) log 10D log 10V

Volume(cm 3)

1 1.0 0.0000 -0.2810 0.52362 2.4 0.3802 0.8596 7.23823 3.8 0.5798 1.4584 28.7310

4 5.2 0.7160 1.8670 73.62235 6.6 0.8195 2.1776 150.53296 8.0 0.9031 2.4283 268.0832


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PART 3


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A) In our daily life,the realtion between two variables is not always in a linear form.For example,the relation between the volume,V,and the diameter,D,in Part 2 above.Plot Vagainst D using suitable scales.

Relationship between V and D

B) When the graph V against D is drawn,the value of m and of n are not easily determined

from the graph.If the non-linear relation is changed to a linear form,a line of best fit can be drawn and the values of the constants and other information can be obtained easily.

a) Reduce the equation V= mD n to a linear form

V=mD n

log10V = log 10 (mD n)log10V=log 10Dn + log 10mlog 10V=n log 10D + log 10m

in which the reduced equation is a linear form of y=Mx + c,

where y = log 10V;M = n;x = log 10D;and C = log 10m. b) Using the data from Part 2,plot the graph and draw the line of best fit.


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Linear Relationship between Log(V) and Log(D)

c) From the graph,find

i) The value of m and of n,thus express V in terms of D,To obtain the value of m we must know the y-intercept,-0.281,because

log 10m= -0.281m=10 -0.281

m=0.5236The value of n can be determined by calculating the slope of the graph,that isn= 2.4283-(-0.281)

0.9031 0 =3Thus volume,V,of a solid sphere can be expressed in terms of D as

V=0.5236(D 3) (6)ii) The volume of the sphere when the diameter is 5 cm,and

log10D=log 105=0.699

With this value 0.699 on the x-axis,we can look up on the linear graph andinterpolate the corresponding value 1.816 on the y-axis.

log10V=1.816V=10 1.816 ~ 65.46 cm 3

iii) The radius of the sphere when the volume is 180 cm 3 log 10V= log 10180 = 2.25527


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With this value 2.25527 on y-axis,we can look up on the linear graph andinterpolate the corresponding value 0.845 on the x-axis.

log 10D= 0.845

r = D = 100.845

2 2 ~3.4992 cm

FURTHER EXPLORATIONa) Compare the equation obtained in Part 3(B) c (i) with the formula of volume of

sphere.Hence,find the value of .

The formula for volume of sphere is given by

Vsphere = 4 r 3

Comparing the formula and Eq.(6),we have

4 D 3 = 0.5236(D 3)3 2

4 D3 = 0.5236(D 3)3 8

1 D3 = 0.5236(D 3)3 2

= 0.52366

= 0.5236 6 = 3.1416

b) Suggest another method to find the value of .


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One simple method is to determine the circumference,C,of a circle with diameter,D.Thecircumference of a circle is the length around it and the associate formula is given by

C = D To measure the circumference,C,of a circle with diameter,D effectively, the shadow of a solid

sphere can be projected on a screen using a bright light source as shown below.Then,thediameter of the casted shadow can be scaled linearly according to the actual diameter of the solidsphere,because the formula shows the linear relationship


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REFLECTION What have you learnt while conducting the project?What moral value did you practice?Expressyour feelings and opinions creatively through the usage of symbols,drawings,lyrics of a song or a

poem.

After spending countless hours,days and night to finish this project and also sacrificingmytime for chatting and movies in this mid year holiday,there are several things that

I can say...Additional Mathematics...From the day I

born...From the day I

was able to holding pencil...From the day I

start learning...And...From the day I heard your name...

I always thought that you will be my greatest obstacle and rival in excelling in my life...But after countless of hours...Countless of days...

Countless of nights...

After sacrificing my precious time just for you...

Sacrificing my play Time..

Sacrificing my Chatting...

Sacrificing my Facebook...

Sacrificing my internet...

Sacrifing my Anime...

Sacrificing my Movies...

I realized something really important in you...

I really love you...You are my real friend...You my partner...You are my soulmate...

I LOVE U ADDITIONAL MATHEMATIC

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