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) $a(w)$ increases as $w$ increases when $w > 0$. $a(w)$ has a minimum value at $w = 5$. $a(w)$ has a maximum value at $w = 5$. $a(w)$ increases as $w$ increases for $0 < w < 5$.

Explanation:

Step1: Analyze the quadratic - function

The function $A(w)=w^{2}+5w$ is a quadratic function in the form $y = ax^{2}+bx + c$, where $a = 1$, $b = 5$, and $c = 0$. The axis - of symmetry of a quadratic function $y=ax^{2}+bx + c$ is given by the formula $w=-\frac{b}{2a}$.

Step2: Calculate the axis - of symmetry

Substitute $a = 1$ and $b = 5$ into the formula $w=-\frac{b}{2a}$. We get $w=-\frac{5}{2\times1}=-\frac{5}{2}$.

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

Answer:

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