Question

The equation  A(w)=w2+5w  represents the relationship between the area in square unitsand 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)ResponsesA(w) increases as w increases when w>0.cap A times w increases as w increases when w is greater than 0.A(w) increases as w increases for 0<w<5.  cap A times w increases as w increases for 0<w<5.A(w) has a minimum value at w=5.  cap A times w has a minimum value at w is equal to 5.A(w) has a maximum value at w=5.

Answer 1

To analyze the equation \( A(w) = w^2 + 5w \), we can derive the nature of the relationship between \( A \) and \( w \).

1. Check the function: The function \( A(w) = w^2 + 5w \) is a quadratic function, which opens upwards (since the coefficient of \( w^2 \) is positive).

2. Find the vertex: The vertex of a quadratic function given by \( A(w) = ax^2 + bx + c \) occurs at \( w = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = 5 \):

   \[ w = -\frac{5}{2 \times 1} = -\frac{5}{2} \]

   Since this vertex is negative, the function is increasing for \( w > 0 \).

3. Evaluate the options:

   • A(w) increases as w increases when w > 0. This statement is true since the function increases for \( w > 0 \).
   • A times w increases as w increases when w is greater than 0. This could be true, but the relationship should be specified since it refers to \( A \) times \( w \) rather than \( A(w) \).
   • A(w) increases as w increases for 0 < w < 5. While this is true, it doesn't capture the entirety of the behavior of the function.
   • A times w has a minimum value at w = 5. This doesn't seem accurate; the function itself doesn't have a minimum at that point.
   • A(w) has a maximum value at w = 5. This is not true; it does not have a maximum since it increases infinitely.

Based on this analysis, the most accurate statement is:

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