1000 FG for help in 45 mins

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My girlfriend is tutoring someone who is enrolled in Precalculus at a local community college. She's spent a number of hours staring at this and working on it and just has a headache and can't get the answers. Her reasoning is included below.




Student Lab

Part A

A particular colony of bacteria has an initial population of 2,000 cells and increases at a rate of 40% per hour. Antibiotic B is taken every 6 hours and has an effectiveness factor of 90%.

1. Determine the bacterial population after three days (72 hours) in a patient who has an initial bacterial population of 2,000 and who is being treated with Antibiotic B. Again, calculate several cycles then find the regression equation.


If we use Rate of growth = g(t)=2000e0.4(t) and rate of death = d(t)=e-.09(t), we could combine them to get f(x)= g(16d(t)). This would equal 2000e0.4(t) x 16e-0.9(t), which equals 333.33e-.5(t). If we plug in our numbers, at 6 hours, (f(6) = 16.59 hours and f(72) = 7.7x1014 hours. This doesn’t seem right.


So, maybe the answer is more simple. Maybe I just use that the growth rate is the formula:

G=2000(1+0.4)t and the rate of death is D=(1-0.1)t.

So, when t = 6, G = 15059.07 and D = .531. I’m not sure what she wants us to do with that.

So, maybe after every 6 hours, 90% of the bacteria die and 10% survive. So, if we just say G=2000(1+0.4)t and when t=6, G = 15059.07. Well, then 90% die. So, G(12) = (.1 x 15059.07)(1+.04)t. Great. The first time, that means that there are fewer bacteria. 1,505.91. However, if you do this twice more, at 18 hours, the bacteria start increasing. So, that doesn’t work.


I’m really not sure how to solve this problem algebraically. Primarily because I have no idea what “effectiveness factor” means, nor do I know what she wants me to do to calculate a regression equation. Does an effectiveness factor of 90% mean the bacteria die gradually, does it mean they die immediately, does it mean they die at the end of 6 hours? Does it mean that they don’t grow sometimes?




2. If less than 50 bacteria cells are considered cured, how many doses will it take for the patient to be considered cured?


See above for issues.


3. Another antibiotic, Antibiotic C, has an effectiveness factor of 99.5%. Explain why antibiotic C cannot cure patients if it is administered every 24 hours. Show the computations that lead to your answer.

Because the rate of growth of the bacteria will be higher than the rate of death. A graph of both rates would never meet.

Part B

In the mid 1300’s, the black plague spread across Europe. Suppose that during a one-year period in a town with population 250,000, 20% of the population died. To model this decline, an exponential decay model would be used, that is y=abx, where b = (1 –r).
3. Given this data, what is the exponential decay model?
Y=a(1- 0.20)x

4. Ignoring the birth rate during this time, how many of the original population would be left after three years?

Y=250,000(1-0.20)3
Y=128,000
5. Suppose this epidemic lasted for 10 years, what is the decade decay rate?
I don’t understand why this wouldn’t be 20%





Part C
The following data table shows the colonial population from 1610 to 1670, where t is the number of years since 1610 and C = C(t) is the population in thousands. The source for the data, the Information Please Almanac, cautions that records from this period are spotty, so the numbers should be considered estimates.

1. Use your calculator to draw the scatter plot. Does the plot indicate that the data might be modeled by an exponential function? Why?

Yes, because the plot appears like a regression line would show exponential growth. (It looks like an exponential graph.)


2. Use exponential regression to construct a model for C. (Round your decimals to the thousandth place)

Y=.704(1.1)x (I did this with my calculator)

3. What is the yearly growth factor? What is the decade growth factor?

I don’t understand her difference.

4. When will it take the population to reach 250 thousand?

From when? From year 1610? I think the population will reach 250,000 in year 1671.61

I don't know much about bacteria and antibiotics. But to me this means... Bacteria grow for 6 hours, then bam, in an instant 90% of the population dies, rinse repeat.

P(6) = 2000*(1+0.4)^6*0.1 =2000*(1.4)^6*0.1 = 1505.9072
P(12) = [P(6)*1.4^6]*0.1=2000*1.4^6*0.1*1.4^6*0.1 = 2000*1.4^12*0.1^2 =1133.8782

P(j) = 2000*1.4^j*0.1^(j/6), j is number of hours, where j = 6n, n is a natural number (it is just a fancy way of saying: only use multiples of 6 for j)

This post was edited by Azrad on Apr 15 2013 09:34pm

P(j) = 2000*1.4^j*0.1^(j/6) < 50

Well I just cheated and plugged in values instead of actually solving that mess, at j = 78 it was just slightly more than 50, so it would take 84 hours (14 treatments) to get less than 50

This post was edited by Azrad on Apr 15 2013 09:55pm

Thanks! I'm not sure if that works. If you plug in 72 (so, 3 days), you get

2000*(1+.04)^(72*.01)^(72/6) which = 2001.52. So, at 72 hours of treatment, you've got more bacteria than you had at 12 hours. It's a similar flaw to what she had been doing. But, question: Why are you raising your exponents like that? (t*r)^(t/6)? Is there a basis for that reasoning?

This post was edited by cd_wizard on Apr 15 2013 09:40pm

your misreading the function, which is partially my fault, let me make it more clear:


This post was edited by Azrad on Apr 15 2013 09:44pm

I'd like to point out that there is a difference between exponential growth and continuous growth

That does work! Thank you! I was stuck with trying to use the equations for exponential growth and decay. My attempt at using the formula you used used 1-.01^t, which had a simple mistake and the bigger error. I really appreciate your assistance.

~The girlfriend

I think all your problems have the same background, and the same form of solutions :

A : +40% every hour and -90% every 6 hours

Just calculate over a period of 6 hours : 1.4^6 * 0.1 ~ 0.753

To sum up, a factor 0.753 is applied to cell population every 6 hours (already solved by Azrad).

To find out how many time it takes to reach less than 50 bacterias, starting from 2000 :

0.753^n < 50/2000
0.753^n < 0.025
n * ln (0.753) < ln (0.025)
n > ln (0.025) / ln (0.753)

Notice that ln(0.753) is a negative value... (as ln(0.025) is).

New antibiotic with 99.5% bacteria death every 24 hours :

1.4^24 * 0.005 ~ 16
16 > 1 : bacteria population still increase (by a factor 16 every 24 hours).

Decade decay rate : in my opinion (I'm not a native english speaker), this means "what is the rate of death on a period of 10 years?".
At 0.8 ber year, this would be 0.8^10

Part C
yearly growth factor = 1.1 (according to your calculations)
decade growth factor = 1.1^10