Part C: Describe a situation involving area in square feet that can be modeled using the function [tex]\( g(x) = 2x^2 - 13x + 6 \)[/tex] so that it relates to the last situation.
Let's consider a situation where you are planting flowers in a rectangular garden where the width of the plot is defined by the function [tex]\( g(x) = 2x^2 - 13x + 6 \)[/tex], and [tex]\( x \)[/tex] represents the length of the garden in feet.
To better understand how the area varies with different garden lengths, let's calculate the area for some specific lengths:
1. For [tex]\( x = 1 \)[/tex] foot:
[tex]\[
g(1) = 2(1)^2 - 13(1) + 6 = -5
\][/tex]
So, the area of the garden plot when the length is 1 foot would be [tex]\(-5\)[/tex] square feet.
2. For [tex]\( x = 2 \)[/tex] feet:
[tex]\[
g(2) = 2(2)^2 - 13(2) + 6 = -12
\][/tex]
So, the area of the garden plot when the length is 2 feet would be [tex]\(-12\)[/tex] square feet.
3. For [tex]\( x = 3 \)[/tex] feet:
[tex]\[
g(3) = 2(3)^2 - 13(3) + 6 = -15
\][/tex]
So, the area of the garden plot when the length is 3 feet would be [tex]\(-15\)[/tex] square feet.
4. For [tex]\( x = 4 \)[/tex] feet:
[tex]\[
g(4) = 2(4)^2 - 13(4) + 6 = -14
\][/tex]
So, the area of the garden plot when the length is 4 feet would be [tex]\(-14\)[/tex] square feet.
5. For [tex]\( x = 5 \)[/tex] feet:
[tex]\[
g(5) = 2(5)^2 - 13(5) + 6 = -9
\][/tex]
So, the area of the garden plot when the length is 5 feet would be [tex]\(-9\)[/tex] square feet.
We notice that the areas are negative which doesn't make practical sense for a physical garden plot, indicating that the model [tex]\( g(x) = 2x^2 - 13x + 6 \)[/tex] may not be appropriate for representing positive garden areas in square feet directly. This might imply that the length values chosen are either inappropriate, or the given function is unsuitable for this specific real-world scenario without further context or adjustments.
However, this analysis clearly demonstrates how the function behaves with different lengths for the garden:
- As the length of the garden changes, the value calculated by the function [tex]\( g(x) \)[/tex] significantly impacts the result.
- Negative values in practical scenarios can denote that the length values or the quadratic model might need reconsidering or further realignment to ensure realistic interpretations.
Thus, it’s important to choose an appropriate model that fits the real-world situation, and this deduction validates the calculated values, illustrating the behavior of the function when used to model the described scenario.
