Algebraic Problems

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 /...