MATH 345 Mathematical Modeling February 14, 2018 1 Modeling A common technique w
ID: 2264586 • Letter: M
Question
MATH 345 Mathematical Modeling February 14, 2018 1 Modeling A common technique when no models are available is to collect data, try to fit curves, and then to treat curves as if they were a model or even an explanation. Discuss. Would you have faith in predictions made from such models? Explain. 2 Exponential Growth I The data given in the table immediately below were recorded as the growth of a colony of bacteria was observed me min tion (p) 10 15 10 20 30 40 50 60 0 80 90 100 50 110 245 1215 2704 6018 13394 29810 (a) Plot this data as function of time. (b) Write an equation that expresses the bacterial population as a function of time. Can you explain why your model is more accurate than other possible models? If your model is valid, can you explain why it fails?Explanation / Answer
1) For exponential growth
dN/dt = r or N = Aert
Taking log we get ln N = ln A + rt i.e. linear model connecting t and ln N.
Hence for the data we can fit a linear regression line using least squares method.
Rewrite in terms of P and t
ln P = 2.303+0.08t
Or P = 10.002e0.08t is the equation expressing relationship between P and t.
b) The graph shows that all points exactly fit on the line. Hence the equation arrived is more accurate.
If deviations are more from the line, then we may get approximate equation only.
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3a) r = the maximum growth rate
K = the maximum capacity that the system can have
N0 = the initial population
K = max capacity and r = maximum growth rate.
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