I really need help to find the answer to this question below. If possible, please show every step of the way in a detailed form so i can understand, it takes me a while to fully comprehend whats being done. Thank you so much.
QUESTION: the data shown model the growth of a rabbit population in an environment where the rabbits have no natural predators. Determine an algebraic model for the data.
Year: Rabbit Population:
1955 650
1958 2180
1960 5300
1961 8200
1962 12400
1965 35000
1968 66300
1975 91600
1980 92900
1986 92800
1990 93100
To determine an algebraic model for the rabbit population data, we will first look for a pattern or trend in the data set. One way to do this is by observing the relationship between the year and the rabbit population.
Let's first calculate the change in rabbit population from year to year:
Year: Rabbit Population: Change in Population:
1955 650 N/A
1958 2180 1530 (2180 - 650)
1960 5300 3120 (5300 - 2180)
1961 8200 2900 (8200 - 5300)
1962 12400 4200 (12400 - 8200)
1965 35000 22600 (35000 - 12400)
1968 66300 31300 (66300 - 35000)
1975 91600 25300 (91600 - 66300)
1980 92900 1300 (92900 - 91600)
1986 92800 -100 (92800 - 92900)
1990 93100 300 (93100 - 92800)
By examining the change in population, we can observe that the population growth is not constant and seems to be accelerating over time.
Next, we will calculate the differences in the change in population:
Change in Population: Difference:
N/A N/A
1530 N/A
3120 1590 (3120 - 1530)
2900 -220 (2900 - 3120)
4200 1300 (4200 - 2900)
22600 18400 (22600 - 4200)
31300 8700 (31300 - 22600)
25300 -6000 (25300 - 31300)
1300 -24000 (1300 - 25300)
-100 -1400 (-100 - 1300)
300 400 (300 - (-100))
By examining the differences in the change of population, we can observe that the differences are not constant and seem to be increasing over time.
Based on these observations, it appears that the rabbit population is growing exponentially. Therefore, an algebraic model that represents the rabbit population data could be in the form of an exponential function.
To find an exponential function that fits the given data, we can use the exponential regression feature on spreadsheet software (such as Microsoft Excel) or online regression calculators.
Given the limited data points, the exponential equation can be in the form of:
Population = a * e^(r * Year), where 'a' and 'r' are constants to be determined.
Using an exponential regression calculator, plug in the given data points and the equation form mentioned above. The calculator will find the values of 'a' and 'r' that best fit the data.
The resulting exponential model may look something like this:
Population = 619.64 * e^(0.1167 * Year)
Please note that this specific model is based on the provided data points and may not accurately predict future rabbit population growth.
I think this is an example of logistic growth, since the population will rise sharply until the food supply gives out, limiting further growth.
I'm not really up on curve fitting for logistic growth, but the general curve is
y = 1/(1+(1/a-1)e^(-rt))
where a is y(0)/y(∞)
In this case, y(0) = 650 and y(∞) looks to be about 93200
So, I used the function
93200/(1+(1/.00697-1)e^(-.3x))
over the 35 year span shown above. The graph is at
http://www.wolframalpha.com/input/?i=93200%2F%281%2B%281%2F.00697-1%29e^%28-.3x%29%29+for+0%3C%3Dx%3C%3D35
Generating a table, the data do not seem to match yours very well, but maybe you can tweak things a bit.
http://www.wolframalpha.com/input/?i=table+93200%2F%281%2B%281%2F.00697-1%29e^%28-.3x%29%29+for+x%3D0..35