ANSWER
The width that will give the maximum area is 27 feet. The maximum area is 1458 square feet.
EXPLANATION
The equation that gives the area is a quadratic function,
[tex]A(x)=x(108-2x)[/tex]To find the width that maximizes the area, we have to find the x-coordinate of the vertex of this parabola. We can observe in the equation that the leading coefficient is -2, so the vertex is the maximum.
First, apply the distributive property to write the equation in standard form,
[tex]A(x)=-2x^2+108x[/tex]The x-coordinate of the vertex of a parabola if the equation is in standard form is,
[tex]\begin{gathered} y=ax^2+bx+c \\ \\ x_{vertex}=\frac{-b}{2a} \end{gathered}[/tex]In this case, b = 108 and a = -2,
[tex]x_{vertex}=\frac{-108}{-2\cdot2}=\frac{108}{4}=27[/tex]Hence, the width that will give the maximum area is 27 feet.
To find the maximum area, we have to find A(27),
[tex]A(27)=27(108-2\cdot27)=27(108-54)=27\cdot54=1458[/tex]Hence, the maximum area is 1458 square feet.