1 - The equation is:
[tex]p(x)=-0.5x^2+6x[/tex]Since it has no variable on exponents and it has a term with x to the second power, this is a quadratic function.
2 - x represents the length of the tomato patch, so if we want the area of the bell pepper patch, we can input x = 8 into the function:
[tex]\begin{gathered} p(8)=-0.5\cdot8^2+6\cdot8 \\ p(8)=-0.5\cdot64+48 \\ p(8)=-32+48 \\ p(8)=16 \end{gathered}[/tex]Thus, the area is 16 square feet.
3 - Since the leading coefficient is negative, the vertex of the parabola gives the maximum value for the function.
The x value of the vertex is:
[tex]\begin{gathered} ax^2+bx+c \\ x_v=-\frac{b}{2a} \end{gathered}[/tex]So, in this case:
[tex]x_v=-\frac{6}{2\cdot(-0.5)}=-\frac{6}{-1}=6[/tex]And to find the value for the area, we need to input that into the function:
[tex]\begin{gathered} p(6)=-0.5\cdot6^2+6\cdot6 \\ p(6)=-0.5\cdot36+36 \\ p(6)=-18+36 \\ p(6)=18 \end{gathered}[/tex]Thus, the maximum area is 18 square feet.
4 - The value for the length of the tomato patch is just the corresponding x for this maximum, which we calculated as the vertex x:
[tex]x_v=6[/tex]Thus, the length of the tomato patch for this maximum value is 6 feet.