Question

read the context below and answer the questions

1. brayden has 24 feet of fence available to build a rectangular fenced in area. if the width of the rectangle is x feet, then the length would be \\( \frac{1}{2}(24 - 2x) \\). a function to find the area, in square feet, of the fenced in rectangle with width x is given by \\( a(x) = \frac{1}{2}x(24 - 2x) \\).

what is an appropriate domain for the function? what would the range be for the domain you chose?

• remember that the given function is quadratic (appropriate for an area)
• the graph of a quadratic function is a parabola which would have a maximum value at the vertex

Explanation:

Step1: Determine domain constraints

For the width \( x \) and length \( \frac{1}{2}(24 - 2x) \), both must be positive. So \( x>0 \) and \( 24 - 2x>0 \) (since length \(=\frac{1}{2}(24 - 2x)>0\) implies \( 24 - 2x>0 \)). Solving \( 24 - 2x>0 \): \( -2x>-24 \) \( \Rightarrow x < 12 \). Thus, domain is \( 0 < x < 12 \) (or \( 0\leq x\leq12 \) if endpoints considered, but length/width positive, so open or closed with \( x = 0,12 \) giving area 0).

Step2: Analyze the function \( A(x)=\frac{1}{2}x(24 - 2x) \)

Simplify \( A(x)=\frac{1}{2}(24x - 2x^{2})=-x^{2}+12x \). This is a quadratic function with \( a=-1<0 \), so it opens downward. The vertex (maximum point) is at \( x =-\frac{b}{2a}=-\frac{12}{2(-1)} = 6 \). At \( x = 6 \), \( A(6)=-(6)^{2}+12(6)=-36 + 72 = 36 \). So the maximum area is 36, and the area is non - negative (since it's an area). For the domain \( 0 < x < 12 \), the range is \( 0 < A(x)\leq36 \) (or \( 0\leq A(x)\leq36 \) if \( x = 0,12 \) included, but those give area 0).

Answer:

• Domain: \( 0 < x < 12 \) (or \( 0\leq x\leq12 \))
• Range: \( 0 < A(x)\leq36 \) (or \( 0\leq A(x)\leq36 \))