Algebra Task

Algebra
Name Course Instructor Date
Part A1- A virus is spreading such that the number of people infected increases by 5% a day. In the first day, 10 people were diagnosed with the virus. How many days will it be before 350 people are diagnosed?
Day 1 10 people
Day 210* 100%+5%= 10*1.05 2-1
Thus the formula is 10*1.05n-1, where n is the number of periods
350= 10* 1.05n-1 35= 1.05n-1(log 35/ log 1.05) +1 = n= 72.87018+1= 73.87 = 74 days
Part BFor one strain of bacteria, each bacterium splits into three every minute. The table shows the number of bacteria present in a particular sample for the first 5 minutes.
Time (minutes)

number of bacteria present

0

3

1

9

2

27

3

81

4

243

5

729

i) Write down an algebraic rule linking the number of bacteria present at a particular time to the number present one minute previously.The number of bacteria at time t is Bt = 3 t+1
* There is a direct relationship between Bt and Bt-1 such that (Bt)-1= Bt-1
* Bt (current period)- 1= Bt-1 (previous one minute)
* Where t is the preset time and t-1 is one previous minute
* The number of bacteria at time t is Bt = 3 t+1
* At time t-1, this is Bt = 3 (t+1)-1 but (t+1)-1= t
Thus the number of bacteria at time t-1 Bt-1=3t
ii) Write down an expression for the number of bacteria present after t minutesAt time 0, there 3 bacteria
At time, 1 minute, there are 9 bacteria
At time, 2 minutes there are 27 bacteria
The replication or growth rate is *3
* Then t= time in minutes
* Initial population = 3 bacteria
* Growth rate =r
* Bt= number of bacteria at time t
The number of bacteria, Bt = (3) ^t+1
iii) Calculate the number of bacteria present after 3 hours. (State any assumptions you make.) 3 hours has 180 minutes
The number of bacteria = (3) ^t+1
Number of bacteria in 180 minutes is 3^ (180+1) = 2.2853204E+86
iv) Calculate the time for the colony to reach over 2.6 million bacteria.2.6 million = (3) ^t+1
So, t= (log 2.6 million/ log 3) = (6.41497/ 0.47712) -1
This is 13.45-1 = 12.45 minutes
Part CUse algebra to extend this model for the growth of bacteria colonies.You could investigate:a) The relationship between the number of bacteria and the size of the colonyBt =kC, where Bt is the number of bacteria, k is a constant and C is the size of the colony
There is a direct relationship between the number of bacteria and the size of the colony such that big colonies tend to have more bacteria and vice versa.
b) Different rates of replication
 

r=2

r=3

r=4

 

V=3

V=3

V=3

0

3

3

3

1

6

9

12

2

12

27

48

3

24

81

192

4

...