Question 1166495: Experts fear that without conservation efforts, tigers could disappear from the wild by 2022. Just one hundred years ago, there were at least 100,000 wild tigers. By 2010, the estimated world tiger population was 3,200. The table below shows the estimated world tiger population for selected years from 1970 through 2010.
Year Wild Tiger Population
1970 37,500
1980 28,000
1990 12,500
2000 6,300
2010 3,200
The polynomial function f(x)=0.76x^3-30x^2-882x+37,807 models the population of wild tigers, where x represents the number of years since 1970.
Using the website https://www.desmos.com/calculator, graph the polynomial using the settings 0 to 40 (step 5) for the x-axis and 0 to 40000 (step 5000) for the y-axis. Do not use commas in any of the numbers. Share the graph by exporting and downloading the image. Include the graph as part of your solution.
Find and interpret the function’s value at 40. Does this overestimate or underestimate the actual data shown in the table? By how much?
Find and interpret the function’s value at 10. Does this overestimate or underestimate the actual data shown in the table? By how much?
Based on the graph alone, what is the lowest population of wild tigers and in what year does this take place?
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! I cannot directly access the external website (Desmos) or include an exported image from it. However, I have generated a representation of the function and the data points based on your specifications, and performed all the required calculations.
## Graphing the Model
The polynomial function is $f(x) = 0.76x^3 - 30x^2 - 882x + 37,807$.
The graph of this function, plotted for the specified ranges ($x$-axis: 0 to 40, $y$-axis: 0 to 40,000), shows a curve that decreases sharply until a minimum point near the end of the domain and then begins to slightly increase.
[Image of wild tiger population model]
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## Analysis of the Function's Value at $x=40$ (Year 2010)
The value $x=40$ represents the year $1970 + 40 = \mathbf{2010}$.
1. **Find the function's value $f(40)$:**
$$f(40) = 0.76(40)^3 - 30(40)^2 - 882(40) + 37,807$$
$$f(40) = 48,640 - 48,000 - 35,280 + 37,807$$
$$f(40) = \mathbf{3,167}$$
2. **Interpretation and Comparison:**
* The model estimates the wild tiger population in 2010 to be **3,167** tigers.
* The actual data for 2010 is **3,200** tigers.
* **Comparison:** $3,167 - 3,200 = -33$.
The function **underestimates** the actual population by **33** tigers.
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## Analysis of the Function’s Value at $x=10$ (Year 1980)
The value $x=10$ represents the year $1970 + 10 = \mathbf{1980}$.
1. **Find the function's value $f(10)$:**
$$f(10) = 0.76(10)^3 - 30(10)^2 - 882(10) + 37,807$$
$$f(10) = 760 - 3,000 - 8,820 + 37,807$$
$$f(10) = \mathbf{26,747}$$
2. **Interpretation and Comparison:**
* The model estimates the wild tiger population in 1980 to be **26,747** tigers.
* The actual data for 1980 is **28,000** tigers.
* **Comparison:** $26,747 - 28,000 = -1,253$.
The function **underestimates** the actual population by **1,253** tigers.
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## Lowest Population Based on the Graph (Model)
The lowest population on the graph for the range $x=[0, 40]$ occurs at the local minimum of the function.
1. **Year of Lowest Population:** The function reaches its minimum at $x \approx 36.79$ years since 1970.
$$1970 + 36.79 \approx \mathbf{2006.79}$$
This occurs during the year **2006** (or 2007).
2. **Lowest Population:** The population at this point is found by calculating $f(36.79)$:
$$f(36.79) \approx \mathbf{2,769}$$
The lowest modeled population of wild tigers is approximately **2,769**, which takes place around the end of the year **2006**.
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