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

Explanation:

Step1: Analyze the function type

The function \( A(w)=w^{2}+5w \) is a quadratic function in the form of \( y = ax^{2}+bx + c \), where \( a = 1 \), \( b = 5 \), \( c = 0 \). Since \( a=1>0 \), the parabola opens upwards.

Step2: Analyze the vertex and increasing/decreasing behavior

For a quadratic function \( y=ax^{2}+bx + c \) with \( a>0 \), the vertex is at \( x=-\frac{b}{2a} \). Here, \( x = -\frac{5}{2\times1}=-\frac{5}{2} \). The function is decreasing when \( x<-\frac{5}{2} \) and increasing when \( x>-\frac{5}{2} \). Since the width \( w>0 \) (because width of a rectangle can't be non - positive in this context), and \( 0>-\frac{5}{2} \), when \( w > 0 \), the function \( A(w) \) is in the increasing part of the parabola. So \( A(w) \) increases as \( w \) increases when \( w>0 \).
Let's check the other options:

• For the option " \( A(w) \) has a minimum value at \( w = 5 \)": The vertex is at \( w=-\frac{5}{2}\), not \( w = 5 \), so this is wrong.
• For the option " \( A(w) \) has a maximum value at \( w = 5 \)": Since \( a>0 \), the parabola opens upwards, so it has a minimum, not a maximum, so this is wrong.
• For the option " \( A(w) \) increases as \( w \) increases for \( 0 < w < 5 \)": Since the function is increasing for \( w>-\frac{5}{2}\), and \( 0>-\frac{5}{2} \), the function is increasing for all \( w > 0 \), not just \( 0 < w < 5 \), so this is wrong.

Answer:

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