Answer:
[tex]\textsf{A)}\quad y=10.4964e^{-0.1413x}[/tex]
B) 2,943
C) 174
No
Step-by-step explanation:
Part A
Population data for endangered panthers have been collected since 2010 and are displayed in the provided scatter plot.
The explanatory (independent) variable represents the number of years elapsed since the start of data collection in 2010 (x-values).
The response (dependent) variable represents the population of endangered panthers in thousands (y-values).
To find the equation for the curve of fit, we need to perform exponential regression analysis on the given data. Using a statistical calculator for this gives the equation for the curve of fit as:
[tex]y=10.4964e^{-0.1413x}[/tex]
where the coefficients are rounded to 4 decimal places.
[tex]\dotfill[/tex]
Part B
To determine the predicted population of endangered panthers in the year 2019, we need to substitute x = 2019 - 2010 = 9 into the model equation and solve for y:
[tex]y=10.4964e^{-0.1413\cdot 9}\\\\y=10.4964e^{-1.2717}\\\\y=2.942714166119...[/tex]
As y represents the population of endangered panthers in thousands, the predicted population of endangered panthers in the year 2019 is approximately 2,943 (rounded to the nearest whole number).
[tex]\dotfill[/tex]
Part C
To determine the predicted population of endangered panthers in the year 2039, substitute x = 2039 - 2010 = 29 into the model equation:
[tex]y=10.4964e^{-0.1413\cdot 29}\\\\y=10.4964e^{-4.0977}\\\\y=0.1743539834...[/tex]
As y represents the population of endangered panthers in thousands, the predicted population of endangered panthers in the year 2039 is approximately 174 (rounded to the nearest whole number).
For this problem, we have used the curve of fit to predict a value of y that is outside the range of the original data. When using values of x outside the range to predict corresponding values of y, this is called extrapolation. These predictions can be unreliable because there is no evidence that the relationship described by the curve of fit is true for all values of x. Beyond the observed data range, various factors such as changes in habitat, human intervention, climate change, and other unforeseen events could significantly influence the population dynamics in ways that are not captured by the model. Therefore, caution should be applied when extrapolating the model to predict population figures far beyond the observed data range.
