Math CentralQuandaries & Queries


Question from sandi, a student:

I have several questions similar to this one and was wondering if you could walk me through this one. I'm totally lost on how to do it.
Paramecia reproduce by splitting in two. In a laboratory flask, a colony of paramecia had an initial population of 500. Each day, the population of the paramecia was counted. The results are as listed.
Time (in days)------Population
1.)Using graphing calculator make a scatter plot of the data in table.
I think I did this part right I set my window at Xmin=0 Xmax=10 Xscl=1 Ymin=0 Ymax=2500 Yscl=100 Xres=1
2.) Determine an exponential equation to represent the population as a function of time without using a graphing calculator.I have no clue how to do this.
3.)Suppose the flask and food supply is large enough to support the trend of the population growth. Estimate the population of the colony when the time is 10 days.


Have you been studying linear regression?


Sandi wrote back

Hi Penny I'm a Distant Ed student and I just got this module. It is entitled Non-Linear Functions. The first question asked me to write a exponential function. The second question asked me what the difference is between exponential growth and decay and this is the third question(very well could be leading up to linear regression). I understand that an exponential equation is written in the form of y=ab^x. Graphing calculators aren't my friend (still trying to figure out how to use one properly, windows really mess me up, but I think I did scatter plot right) Originally I was going to do the following y=ab^t/p (y being the future amount, a being the initial amount(500), b being type of growth(2 cause it says splitting in two), p being period of growth(24hours) and t being time?not sure what to put or if this should be the24hours. For the time I was going to turn the days into hours so I'd have 24,48,72,96,120,144,168,192,216,240. So y=500(20^? not sure what I'm doing at this point. Is anything I'm doing making sense or should this question be handled a totally different way cause the t and p part have me stumped and second guessing everything.
Any help would be greatly appreciated. Some of the other questions I have are similar to this one so I'm hoping I can use this as a model.


My first thought was much like yours. My thought was to use the equation

y = a bkx,

with b = 2 or b = 10 or if you know about natural logarithms, b = e. Now take the logarithm of both sides and the equation becomes

log(y) = log(a bkx) = log(a) + k log(b) x.

If you now let Y = log(y), A = log(a) and m = k log(b) then you can see this is a linear equation

Y = A + m x.

This is a standard technique when you expect the relationship is exponential. Take the logarithm of each value in the y-column, in your case the population column, to get Y-values and construct a scatter plot of Y against x (time in your problem). If the original relationship is exponential this scatter plot will look linear. Use some technique (linear regression) to fit a linear equation Y = A + m x and then, depending on the value of b you picked, recover the values of a and k.

This looks too complex for what seems to be expected of you in this problem so I went back and looked at your data, in particular the population values. What I noticed is that

600 = 500 × 6/5
720 = 600 × 6/5 = 500 × (6/5)2
864 = 720 × 6/5 = 500 × (6/5)3
and so on.

Can you complete the problem now?

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