To answer this question we first, have to put the length of the rectangle in terms of x. We know that the total area is 150 ft², and that the area of a rectangle is given by:
[tex]A=\text{width}\cdot\text{length.}[/tex]Solving the above equation for the length, and substituting width=x, and A=150, we get:
[tex]\text{length}=\frac{150}{x}\text{.}[/tex]Now, the length of the interior rectangle is:
[tex]\frac{150}{x}-8,[/tex]and the width of the interior triangle is:
[tex]x-4.[/tex]Therefore, the area of the interior triangle is given by the following expression:
[tex](x-4)(\frac{150}{x}-8)\text{.}[/tex]Now, to determine the domain, we know that the sides of the interior rectangle must fulfill the following inequalities:
[tex]\begin{gathered} x-4>0, \\ \frac{150}{x}-8>0. \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} x>4, \\ \frac{150}{x}>8, \\ 150>8x, \\ \frac{150}{8}>x\text{.} \end{gathered}[/tex]Answer:
Area as a function of x
[tex](x-4)(\frac{150}{x}-8)\text{.}[/tex]Domain:
[tex](4,\frac{75}{4})\text{.}[/tex]