add math selangor 4
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pg. 1
PART 1
a) Write a history on logarithm.
History of Logarithms
From Napier to Euler
The method of logarithms was publicly
propounded byJohn Napierin 1614, in a
book titled Mirifici Logarithmorum Canonis
Descriptio(Description of the Wonderful
Rule of Logarithms).Joost
Brgiindependently invented logarithms
but published six years after Napier.
Johannes Kepler, who used logarithm
tables extensively to compile
his Ephemeris and therefore dedicated it
to Napier, remarked:
...the accent in calculation led Justus
Byrgius [Joost Brgi] on the way to these
very logarithms many years before
Napier's system appeared; but ...instead of rearing up his child for the public benefit he
deserted it in the birth.
Johannes Kepler, Rudolphine Tables (1627)
By repeated subtractions Napier calculated (1 107)L forL ranging from 1 to 100. The
result forL=100 is approximately0.99999 = 1 105. Napier then calculated the
products of these numbers with 107(1 105)L forL from 1 to 50, and did similarly
with0.9998 (1 105)20 and 0.9 0.99520. These computations, which occupied 20
years, allowed him to give, for any numberNfrom 5 to 10 million, the numberL that
solves the equation
Napier first called L an "artificial number", but later introduced the word "logarithm"to
mean a number that indicates a ratio: (logos) meaning proportion,
and (arithmos) meaning number. In modern notation, the relation to natural
logarithms is
T
http://en.wikipedia.org/wiki/John_Napierhttp://en.wikipedia.org/wiki/John_Napierhttp://en.wikipedia.org/wiki/John_Napierhttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Logoshttp://en.wikipedia.org/wiki/Logoshttp://en.wikipedia.org/wiki/Logoshttp://en.wikipedia.org/wiki/File:John_Napier.jpghttp://en.wikipedia.org/wiki/File:John_Napier.jpghttp://en.wikipedia.org/wiki/File:John_Napier.jpghttp://en.wikipedia.org/wiki/File:John_Napier.jpghttp://en.wikipedia.org/wiki/Logoshttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/John_Napier -
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pg. 2
where the very close approximation corresponds to the observation that
The invention was quickly and widely met with acclaim. The works
ofBonaventura Cavalieri(Italy),Edmund Wingate(France), Xue Fengzuo(China), andJohannes Kepler's Chilias logarithmorum (Germany) helped spread
the concept further.
In 1647Grgoire de Saint-Vincentrelated logarithms to the quadrature of the
hyperbola, by pointing out that the area f(t) under the hyperbola fromx=
1 tox= tsatisfies
The natural logarithm was first described byNicholas Mercatorin his
work Logarithmotechnia published in 1668, although the mathematics teacher JohnSpeidell had already in 1619 compiled a table on the natural logarithm. Around
1730,Leonhard Eulerdefined the exponential function and the natural logarithm by
Euler also showed that the two functions are inverse to one another.
http://en.wikipedia.org/wiki/Bonaventura_Cavalierihttp://en.wikipedia.org/wiki/Bonaventura_Cavalierihttp://en.wikipedia.org/wiki/Bonaventura_Cavalierihttp://en.wikipedia.org/wiki/Edmund_Wingatehttp://en.wikipedia.org/wiki/Edmund_Wingatehttp://en.wikipedia.org/wiki/Edmund_Wingatehttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Gr%C3%A9goire_de_Saint-Vincenthttp://en.wikipedia.org/wiki/Gr%C3%A9goire_de_Saint-Vincenthttp://en.wikipedia.org/wiki/Gr%C3%A9goire_de_Saint-Vincenthttp://en.wikipedia.org/wiki/Nicholas_Mercatorhttp://en.wikipedia.org/wiki/Nicholas_Mercatorhttp://en.wikipedia.org/wiki/Nicholas_Mercatorhttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Nicholas_Mercatorhttp://en.wikipedia.org/wiki/Gr%C3%A9goire_de_Saint-Vincenthttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Edmund_Wingatehttp://en.wikipedia.org/wiki/Bonaventura_Cavalieri -
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pg. 4
maximum of the likelihood function occurs at the same parameter-value as a maximum of the
logarithm of the likelihood (the "log likelihood"), because the logarithm is an increasing
function. The log-likelihood is easier to maximize, especially for the multiplied likelihoods
forindependentrandom variables.
Benford's lawdescribes the occurrence of digits in manydata sets, such as heights of buildings.According to Benford's law, the probability that the first decimal-digit of an item in the data
sample is d(from 1 to 9) equals log10(d+ 1) log10(d), regardless of the unit of measurement.
Thus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc.
Auditors examine deviations from Benford's law to detect fraudulent accounting.
2. Fractals
The Sierpinski triangle (at the right) is constructed by repeatedly replacingequilateral
trianglesby three smaller ones.
Logarithms occur in definitions of thedimensionoffractals. Fractals are geometric objects that
areself-similar: small parts reproduce, at least roughly, the entire global structure. TheSierpinski
triangle(pictured) can be covered by three copies of itself, each having sides half the original
length. This makes theHausdorff dimensionof this structure log(3)/log(2) 1.58. Another
logarithm-based notion of dimension is obtained bycounting the number of boxesneeded to
cover the fractal in question.
http://en.wikipedia.org/wiki/Independence_(probability)http://en.wikipedia.org/wiki/Independence_(probability)http://en.wikipedia.org/wiki/Independence_(probability)http://en.wikipedia.org/wiki/Benford%27s_lawhttp://en.wikipedia.org/wiki/Benford%27s_lawhttp://en.wikipedia.org/wiki/Data_sethttp://en.wikipedia.org/wiki/Data_sethttp://en.wikipedia.org/wiki/Data_sethttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Fractalhttp://en.wikipedia.org/wiki/Fractalhttp://en.wikipedia.org/wiki/Fractalhttp://en.wikipedia.org/wiki/Self-similarityhttp://en.wikipedia.org/wiki/Self-similarityhttp://en.wikipedia.org/wiki/Self-similarityhttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Hausdorff_dimensionhttp://en.wikipedia.org/wiki/Hausdorff_dimensionhttp://en.wikipedia.org/wiki/Hausdorff_dimensionhttp://en.wikipedia.org/wiki/Box-counting_dimensionhttp://en.wikipedia.org/wiki/Box-counting_dimensionhttp://en.wikipedia.org/wiki/Box-counting_dimensionhttp://en.wikipedia.org/wiki/File:Sierpinski_dimension.svghttp://en.wikipedia.org/wiki/Box-counting_dimensionhttp://en.wikipedia.org/wiki/Hausdorff_dimensionhttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Self-similarityhttp://en.wikipedia.org/wiki/Fractalhttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Data_sethttp://en.wikipedia.org/wiki/Benford%27s_lawhttp://en.wikipedia.org/wiki/Independence_(probability) -
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pg. 5
PART 2
The volume, V, in cm3, of a solid sphere and its diameter, D, in cm, are related by the
equation , where m and n are constants.
Find the value of m and n by conducting the activities below.
I. Choose 6 different spheres with diameters between 1cm to 8cm. The diameterof the 6 spheres using a pair of vernier calipers.
II. Find the volume of each sphere using water displacement method.III. Tabulate the values of diameter, D, in cm and its corresponding volume, V, cm3.
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pg. 6
Find the volume of sphere using water displacement method.
A method of finding the volume of a sphere with minimal calculations is to use the WaterDisplacement Method:
1. Fill a beaker or graduated cylinder with enough water to completely immerse thesphere in.
2. Record the baseline initial measurement3. Drop the sphere in4. Record final measurement5. Subtract the initial volume from the final volume ~ this is the volume of the
sphere!
Value of diameter,D and Volume
Diameter,D ( Volume, V (
D1 = 1.0 V1= 0.5
D2 =2.8 V2= 11.5
D3 =4.0 V3= 34.0
D4 =5.2 V4= 74.0
D5 =6.6 V5= 151.0
D6 =7.8 V6= 250.0
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pg. 7
Diameter,D ( Volume, V (
D1 = 1.0 V1= 0.5
D2 =2.8 V2= 11.5
D3 =4.0 V3= 34.0
D4 =5.2 V4= 74.0D5 =6.6 V5= 151.0
D6 =7.8 V6= 250.0
We can solve by simultaneous method
Substitute the values in the equation
We obtain,
----------(1)
----------(2)
-----------(3)Substitute (3) into (2)
D2 = 2.8 V2= 11.5
D5 =6.6 V5= 151.0
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pg. 8
-----------(4)
Substitute (4) into (3)
Therefore, and
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pg. 9
PART 3
3(A)
D v
1.0 0.5
2.8 11.5
4.0 34.0
5.2 74.0
6.6 151.0
7.8 250.0
y = 0.505x3.025
0
50
100
150
200
250
300
0 1 2 3 4 5 6 7 8 9
Volume,V
Diameter, D
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pg. 10
3(B)
y = 3.025x - 0.2967
-0.5
0
0.5
1
1.5
2
2.5
3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
logV
Diameter, D
log D log V
0 -0.30103
0.447158 1.060698
0.60206 1.531479
0.716003 1.869232
0.819544 2.178977
0.892095 2.39794
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pg. 11
3c) From the graph, find
1. The value of m and of n, thus express V in terms of D.
(nearest whole number)
y = 3.025x - 0.2967
-0.5
0
0.5
1
1.5
2
2.5
3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
logV
Diameter, D
log D log V
0 -0.30103
0.447158 1.060698
0.60206 1.531479
0.716003 1.869232
0.819544 2.178977
0.892095 2.39794
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pg. 12
2. Volume of the sphere when diameter is 5cm
Since graph is logV against logD, we need to transfer, D=5cm int0
logD=log5=0.6989
We get
3. The radius of the sphere when the volume is
Change to logv=log180=2.25,
From the graph, we get
y = 3.025x - 0.2967
-0.5
0
0.5
1
1.5
2
2.5
3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
logV
Diameter, D
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pg. 13
FURTHER EXPLORATION
a) -------(1)
------------(2)
(1)=(2)
D
-------------------cancel on both sides
b) Another method to find value of is using Monte Carlo simulation or
Archimedes method of Exhaustion
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pg. 14
REFLECTION
Symbols used in this project using Microsoft word equation insert tool really help me so
much here are some of the symbol I use.
I really learn how to use Microsoft excel and word to do graph, insert equation and a lot
more.
y = 0.505x3.025
0
50
100
150
200
250
300
0 2 4 6 8 10
Vo
lume,V
Diameter, D
y = 3.025x - 0.2967
-0.5
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1
logV
Diameter, D
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pg 15
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