To analyze the equation \( A(w) = w^2 + 5w \), we can understand how the area \( A \) changes with respect to the width \( w \).
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The equation \( A(w) = w^2 + 5w \) is a quadratic function that opens upwards (since the coefficient of \( w^2 \) is positive).
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To find the vertex of this quadratic, which represents the maximum or minimum point, we can use the formula for the vertex \( w = -\frac{b}{2a} \), where \( a = 1 \) and \( b = 5 \).
\[ w = -\frac{5}{2 \times 1} = -\frac{5}{2} \]
This tells us that the vertex (minimum) occurs at \( w = -2.5 \), which is outside the range of positive widths.
- Therefore, the function \( A(w) \) does not have a maximum; it will keep increasing as \( w \) increases.
To specifically address the responses:
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A(w) increases as w increases when w > 0. — This statement is true. The function is always increasing for positive values of \( w \).
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A(w) has a maximum value at w = 5. — This statement is false. The function does not have a maximum.
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A(w) has a minimum value at w = 5. — This statement is false. The minimum occurs at \( w = -2.5 \).
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A(w) increases as w increases for \( 0 < w < 5 \). — This statement is also true but doesn't capture the behavior for \( w > 5 \).
The best choice among the given options is:
A(w) increases as w increases when w > 0.