Question 1166495
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**.