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 given data, we need to analyze the relationship between the year and the rabbit population. The goal is to find a mathematical equation that represents this relationship.
Step 1: Analyzing the difference in population growth between consecutive years
First, let's calculate the differences in the rabbit population between consecutive years:
1958 - 1955 = 3 years, population difference = 2180 - 650 = 1530
1960 - 1958 = 2 years, population difference = 5300 - 2180 = 3120
1961 - 1960 = 1 year, population difference = 8200 - 5300 = 2900
1962 - 1961 = 1 year, population difference = 12400 - 8200 = 4200
1965 - 1962 = 3 years, population difference = 35000 - 12400 = 22600
1968 - 1965 = 3 years, population difference = 66300 - 35000 = 31300
1975 - 1968 = 7 years, population difference = 91600 - 66300 = 25300
1980 - 1975 = 5 years, population difference = 92900 - 91600 = 1300
1986 - 1980 = 6 years, population difference = 92800 - 92900 = -100
1990 - 1986 = 4 years, population difference = 93100 - 92800 = 300
Observation: It seems that the population difference between consecutive years is not consistent.
Step 2: Analyzing the growth rate of the rabbit population
Instead of focusing on the absolute population difference between consecutive years, let's examine the growth rate, which is the relative increase from one year to the next.
To calculate the growth rate, we divide the population difference by the number of years between the two data points:
(2180 - 650) / (1958 - 1955) = 1530 / 3 = 510
(5300 - 2180) / (1960 - 1958) = 3120 / 2 = 1560
(8200 - 5300) / (1961 - 1960) = 2900 / 1 = 2900
(12400 - 8200) / (1962 - 1961) = 4200 / 1 = 4200
(35000 - 12400) / (1965 - 1962) = 22600 / 3 ≈ 7533.3
(66300 - 35000) / (1968 - 1965) = 31300 / 3 ≈ 10433.3
(91600 - 66300) / (1975 - 1968) = 25300 / 7 ≈ 3614.3
(92900 - 91600) / (1980 - 1975) = 1300 / 5 = 260
(92800 - 92900) / (1986 - 1980) = -100 / 6 ≈ -16.7
(93100 - 92800) / (1990 - 1986) = 300 / 4 = 75
Observation: The growth rates are not constant; they vary from year to year.
Step 3: Looking for a pattern
Based on the growth rates calculated above, we need to identify a pattern that can be expressed algebraically.
Observation: It appears that the growth rate is getting smaller as the population increases.
Step 4: Building a model
To create an algebraic model, we can use a combination of exponential and polynomial functions to capture the growth pattern observed.
Based on the data and the pattern observed, a possible algebraic model for the rabbit population could be:
Population(year) = a * (b^year) + c * (year^2)
Where:
- Population(year) is the rabbit population at a specific year.
- year represents the number of years after the first recorded year (1955).
- a, b, and c are coefficients that need to be found.
Step 5: Using the data to find the values of the coefficients
To find the values of the coefficients, we can use the given data points and solve a system of equations. We can substitute the year and population values into the model and solve for the coefficients.
Using the first three data points (1955, 1958, 2180):
650 = a * (b^0) + c * (0^2)
2180 = a * (b^3) + c * (3^2)
5300 = a * (b^5) + c * (5^2)
Simplifying the equations:
650 = a + c * 0
2180 = a * b^3 + 9c
5300 = a * b^5 + 25c
This system of equations can be solved to find the values of a, b, and c. By solving this system, we can obtain the coefficients needed for the algebraic model that represents the rabbit population growth.