Part 1

a) John Napier (15501617), the inventor of logarithms. The method of logarithms was publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms).Joost Brgi independently invented logarithms but published six years after Napier.

b) Biology

I start a biology experiment with 5 000 000 cells and 45% of the cells are decaying every 10 minutes,how long will it take to have less than 1 000 cells?

c) Application of Logarithmic Function:0 min , number of cells, T1 = 5 000 000

After 10 mins, number of cells,

T2=5 000 000 5 000 000 * 45% = 2 750 000

After 20 mins, number of cells,

T3= 2 750 000 2 750 000 * 45% = 1 512 500

a= 5 000 000, r = 0.55, Tn < 1 000

5 000 000(0.55n-1) < 1 000

(n-1)log 0.55 < log (1000/5 000 000)

(n-1) < 14.24668125

n < 15.24668125

n=15

Time taken = (15-1) 10 =140 mins

PART 2

a)iii)SphereDiameter(cm)Volume(round off to the nearest 0.1cm3)

11.00.5

22.47.2

33.828.7

45.273.6

56.6150.6

68268

b) 268=m(8n)1 0.5=m(1.0n)..2

268=[ 8 ] n0.5 1.0

n= 3

m= 0.5236

PART 3

(A)

(B) a) Reduce the equation V=mDn to a linear form

V=mDn

log10 V= log10(mDn)

log10 V= log10 Dn +log10 m

log10 V= n log10 D+ log10 m

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

where y=log10 V;M = n; x= log10 D; and C= log10 m

b) Using the data from Part 2, plot the graph and draw the line of best fit

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

log10 m = -0.281

m= 10-0.281

m=0.5236

The value of n can be determined by calculating the slope of the graph, that is

n= 2.4283-(-0.281) = 3 0.9031 0

Thus volume,V,of a solid sphere can be expressed in terms of D as

V=0.5236(D3)

ii)the volume of the sphere when the diameter is 5cm,and

log10 D= log10 5= 0.699

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

log10 V = 1.816

V= 101.816 ~ 65.46 cm3

iii)the radius of the sphere when the volume is 180 cm3

log10 V = log10 180 = 2.25527

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

log10 D = 0.845 r = D=100.845 ~ 3.4992 2 2

FURTHER EXPLORATION

a) Compare the equation obtained in Part 3 (B) c (i) with formula of volume of sphere.Hence,find the value of 22 7

b) Suggest another method to find the value of 22 7

REFLECTION

After by spending countless hours, days and night to finish this project and also sacrificing our time for chatting and movies in this few weeks, there are several things that we want tosayAdditional Mathematics,The killer subject,But when we study hard,It was so easy to understandAdditional Mathematics,You look so interesting,So unique from the other subject,Thats why we like you so muchAfter sacrificing our precious time,Spirit and energy for this project,And now,We realized something important from it!

We really love Additional Mathematics,Additional Mathematics,You are our real friend,You are our family,And you are our life

WE LOVE ADDITIONAL MATHEMATICS!