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The equation A(w)=w2+5w represents the relationship between the area in square unitsand the width of a rectangle whose length i...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.
A. $A(w)$ increases as [tex]$w$[/tex] increases for $0 \ \textless \ w \ \textless \ 5$.
B. $A(w)$ increases as [tex]$w$[/tex] increases when $w \ \textgreater \ 0$.
C. $A(w)$ has a minimum value at [tex]$w = 5$[/tex].
D. $A(w)$ has a maximum value at $w = 5$.
Select the sentence that describes an accurate relationship between A and w.
A. $A(w)$ increases as [tex]$w$[/tex] increases for $0 \ \textless \ w \ \textless \ 5$.
B. $A(w)$ increases as [tex]$w$[/tex] increases when $w \ \textgreater \ 0$.
C. $A(w)$ has a minimum value at [tex]$w = 5$[/tex].
D. $A(w)$ has a maximum value at $w = 5$.
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Answered by
GPT-4o mini
To analyze the function \( A(w) = w^2 + 5w \), we can determine its behavior over different intervals and identify any minimum or maximum values it may have.
1. **Determine the nature of the quadratic function**:
The equation \( A(w) = w^2 + 5w \) is a quadratic function in the standard form \( A(w) = aw^2 + bw + c \), where \( a = 1 \), \( b = 5 \), and \( c = 0 \). Since \( a > 0 \), this function opens upwards, indicating that it has a minimum point and no maximum values.
2. **Finding the vertex (minimum value)**:
The vertex of a parabola represented by \( A(w) = aw^2 + bw + c \) occurs at \( w = -\frac{b}{2a} \).
Here,
\[
w = -\frac{5}{2 \cdot 1} = -\frac{5}{2} = -2.5.
\]
Since the vertex is at \( w = -2.5 \), and this value is not in the interval \( w > 0 \), it indicates the function keeps increasing for all \( w > 0 \).
3. **Behavior of \( A(w) \)**:
For \( w > 0 \), since \( A(w) \) is a parabola that opens upwards, it will increase for \( w > -2.5 \) (which is always true for \( w > 0 \)). Therefore, \( A(w) \) increases as \( w \) increases for \( w > 0 \).
4. **Interval analysis**:
In the interval \( 0 < w < 5 \), since the function keeps increasing, we can confirm that \( A(w) \) increases in that range as well.
5. **Conclusion on options**:
- Option A states that \( A(w) \) increases as \( w \) increases for \( 0 < w < 5 \): This is true.
- Option B states that \( A(w) \) increases as \( w \) increases when \( w > 0 \): This is also true.
- Option C states that \( A(w) \) has a minimum value at \( w = 5\): This is false; it has its minimum at \( w = -2.5\).
- Option D states that \( A(w) \) has a maximum value at \( w = 5\): This is also false; the function has no maximum as it continues to increase indefinitely.
Since both Options A and B are accurate, the most comprehensive and general statement about the behavior of \( A(w) \) is option B:
**B. \( A(w) \) increases as \( w \) increases when \( w > 0 \).**
1. **Determine the nature of the quadratic function**:
The equation \( A(w) = w^2 + 5w \) is a quadratic function in the standard form \( A(w) = aw^2 + bw + c \), where \( a = 1 \), \( b = 5 \), and \( c = 0 \). Since \( a > 0 \), this function opens upwards, indicating that it has a minimum point and no maximum values.
2. **Finding the vertex (minimum value)**:
The vertex of a parabola represented by \( A(w) = aw^2 + bw + c \) occurs at \( w = -\frac{b}{2a} \).
Here,
\[
w = -\frac{5}{2 \cdot 1} = -\frac{5}{2} = -2.5.
\]
Since the vertex is at \( w = -2.5 \), and this value is not in the interval \( w > 0 \), it indicates the function keeps increasing for all \( w > 0 \).
3. **Behavior of \( A(w) \)**:
For \( w > 0 \), since \( A(w) \) is a parabola that opens upwards, it will increase for \( w > -2.5 \) (which is always true for \( w > 0 \)). Therefore, \( A(w) \) increases as \( w \) increases for \( w > 0 \).
4. **Interval analysis**:
In the interval \( 0 < w < 5 \), since the function keeps increasing, we can confirm that \( A(w) \) increases in that range as well.
5. **Conclusion on options**:
- Option A states that \( A(w) \) increases as \( w \) increases for \( 0 < w < 5 \): This is true.
- Option B states that \( A(w) \) increases as \( w \) increases when \( w > 0 \): This is also true.
- Option C states that \( A(w) \) has a minimum value at \( w = 5\): This is false; it has its minimum at \( w = -2.5\).
- Option D states that \( A(w) \) has a maximum value at \( w = 5\): This is also false; the function has no maximum as it continues to increase indefinitely.
Since both Options A and B are accurate, the most comprehensive and general statement about the behavior of \( A(w) \) is option B:
**B. \( A(w) \) increases as \( w \) increases when \( w > 0 \).**
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