Question

The equation A(w)=w2+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)
Responses

A(w)
has a maximum value at w=5
.
cap A times w has a maximum value at w is equal to 5.

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)
increases as w
increases when w>0
.

Answer 1

To analyze the function \( A(w) = w^2 + 5w \), we can start by rewriting it in a more familiar form. This is a quadratic function that opens upwards (since the coefficient of \( w^2 \) is positive).

1. **Finding the Vertex**: The vertex of a quadratic function in the form \( A(w) = aw^2 + bw + c \) occurs at \( w = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = 5 \).

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

Since this vertex is negative, the function will increase as \( w \) increases from 0 to infinity.

2. **Behavior of the Function**:
- As \( w \) increases, \( A(w) \) will also keep increasing for \( w > 0 \).

3. **Options Evaluation**:
- **A(w) has a maximum value at w=5**: This is incorrect; it has a minimum value at the vertex (which we don't reach because the vertex is negative).
- **A times w has a maximum value at w=5**: This seems based on a misunderstanding; not applicable.
- **A(w) increases as w increases for 0<w<5**: While it is true that \( A(w) \) is increasing for this range, it doesn’t stop increasing as \( w \) goes beyond 5.
- **A times w increases as w increases for 0<w<5**: Same as previous point but wrongly rephrased.
- **A(w) has a minimum value at w=5**: This is incorrect for the same reason as the first point.
- **A(w) increases as w increases when w>0**: This statement is true.

The most accurate choice is:

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