Algebraic Problems
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Please part C only !!
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Part A
1- 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?
Part B
For 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.
ii) Write down an expression for the number of bacteria present after t
minutes
iii) Calculate the number of bacteria present after 3 hours. (State any
assumptions you make.)
iv) Calculate the time for the colony to reach over 2.6 million bacteria.
Part C
Use 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
colony
b) Different rates of replication
c) Colonies of different sizes at the start
d) Effect of growth limiting factors (such as build-up of waste products,
competition for space)
Explain all the steps of your analysis and state any assumptions you make in
constructing your model. Give references for any data you use.
kindly to do part C only
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 were diagnosed
Day 2, 10* 1.05= 10*(1.05 2-1) = 10.25
Day 3, 10* 1.05= 10*1.05 3-1) = 11.025
D= 10*1.05t-1, where t is the number of periods
350= 10* 1.05n-1 35= 1.05n-11.05t-1 = 35
Then t-1 is log 25/ log 1.05= 72.87
T-1= 72.87 days and t= 73.87
In≈ 74 days, 350 people will be diagnosed.
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.
* Number of bacteria at time is Bt = 3 t+1
* Number present one minute previously Bt-1 = 3t
There is a direct relationship between the number of bacteria and time in an exponential growth model. The number of bacteria is equal to the 3 to the power of t+1 where t is the time in minutes. For instance to determine the number of bacteria at 4 minutes, then t+1 is 4+1=5 and the number of bacteria is 3^5= 243
ii) Write down an expression for the number of bacteria present after t minutes
* The number of bacteria, Bt = 3 t+1 and is also equal to Bt = 3*3t
Bt is the number of bacteria at t minutes
3 is a constant that represents the initial population at time 0 minutes
Growth rate is 3t such that at 1 minute the growth rate is 31
At 1 minute the number of bacteria is 3*3t= 3*31= 9
iii) Calculate the number of bacteria present after 3 hours. (State any assumptions you make.)
1 hour = 60 minutes
3 hours = 180 minutes
The number of bacteria when t is 180 minutes is Bt = 3*3t =3*3180
* Bt = 3* 7.61773E+85 = 2.2853E+86
iv) Calculate the time for the colony to reach over 2.6 million bacteria.Bt = 3*3t
Bt =2.6 million
2.6 million = 3*3t
Using logs then, log 2.6 million- log 3 = t log 3
Log 2.6 million- log 3 =5.9379
Log 3= 0.4771
Then (log (2.6 million/ - log 3 /...
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