SOLUTION: MAT 145: Topics In Contemporary Math 20: Logistic Growth 1) The population of fuzzy quadrupeds increases by 50%) every month after they are introduced to a new regio

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Question 1193432: MAT 145: Topics In Contemporary Math
20: Logistic Growth

1) The population of fuzzy quadrupeds increases by 50%) every month after they are
introduced to a new region. Suppose 30 fuzzy quadrupeds are accidentally introduced to
Springfield when a family visited the region to which fuzzy quadrupeds are native.
However, Springfield only has a carrying capacity of 70 fuzzy quadrupeds. How many
fuzzy quadrupeds would there be for each of the first five months after they were
introduced to the new region?
Month Formula Population size
0 Beginning population
1
2
3
4
5
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


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(Answered the other day for (perhaps) a different student...)

Solution copied here.

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Here is a logistic growth formula:

f%28x%29=L%2F%281%2Ba%2Ae%5E%28-bt%29%29

L is the carrying capacity, given as 70.

a and b are constants to be determined from the given information.

The initial population, f(0), is 30:

f%280%29=70%2F%281%2Ba%2Ae%5E0%29=70%2F%281%2Ba%29=30
70=30%281%2Ba%29=30%2B30a
40=30a
a=40%2F30=4%2F3

The population increases by 50% each month, so the population after one month is 1.5(30)=45:

45=70%2F%281%2B%284%2F3%29e%5E%28%28-b%281%29%29%29%29
70=45%281%2B%284%2F3%29e%5E%28-b%29%29
70=45%2B60e%5E%28-b%29
25=60e%5E%28-b%29
e%5E%28-b%29=25%2F60=5%2F12
-b=ln%285%2F12%29 =-0.87547 to 5 decimal places
b=0.87547

f%28n%29=70%2F%281%2B%284%2F3%29e%5E%28%28-0.87547%29n%29%29

That function gives the following populations after n months to fill your chart:
  months   population  (rounded)
 -------------------------------
    0      30             30
    1      45             45
    2      56.842         57
    3      63.842         64
    4      67.296         67
    5      68.847         69
    .
    .
    .
   10      69.985         70