Question

the equation $a(w)=w^{2}+5w$ represents the relationship between the area in square units and the width of a rectangle whose length is 5 units longer than its width. select the sentence that describes an accurate relationship between $a$ and $w$. (1 point) $a(w)$ has a minimum value at $w = 5$. $a(w)$ increases as $w$ increases for $0 < w < 5$. $a(w)$ increases as $w$ increases when $w>0$. $a(w)$ has a maximum value at $w = 5$.

Explanation:

Step1: Identify the function type

$A(w)=w^{2}+5w$ is a quadratic function in the form $y = ax^{2}+bx + c$ where $a = 1$, $b=5$ and $c = 0$.

Step2: Find the vertex - x - coordinate

The x - coordinate of the vertex of a quadratic function $y=ax^{2}+bx + c$ is given by $x=-\frac{b}{2a}$. For $A(w)=w^{2}+5w$, we have $w=-\frac{5}{2\times1}=-\frac{5}{2}$.

Step3: Analyze the behavior of the function

Since $a = 1>0$, the parabola opens upward. For $w>-\frac{5}{2}$, the function $A(w)$ is increasing. When $w>0$ (width cannot be negative in the context of a rectangle), as $w$ increases, $A(w)$ increases.

Answer:

$A(w)$ increases as $w$ increases when $w > 0$.