Question

Area as a function of x = Domain of the function of the area =

Answer 1

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:

`A=\text{width}\cdot\text{length.}`

Solving the above equation for the length, and substituting width=x, and A=150, we get:

`\text{length}=(150)/(x)\text{.}`

Now, the length of the interior rectangle is:

`(150)/(x)-8,`

and the width of the interior triangle is:

`x-4.`

Therefore, the area of the interior triangle is given by the following expression:

`(x-4)((150)/(x)-8)\text{.}`

Now, to determine the domain, we know that the sides of the interior rectangle must fulfill the following inequalities:

`\begin{gathered} x-4>0, \\ (150)/(x)-8>0. \end{gathered}`

Therefore,

`\begin{gathered} x>4, \\ (150)/(x)>8, \\ 150>8x, \\ (150)/(8)>x\text{.} \end{gathered}`

Answer:

Area as a function of x

`(x-4)((150)/(x)-8)\text{.}`

Domain:

`(4,(75)/(4))\text{.}`