The equation A(w)=w2+5w

Question

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 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
.
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 maximum value at w=5
.
cap A times w has a maximum value at w is equal to 5.
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Answer 1

To analyze the function \( A(w) = w^2 + 5w \), we can rewrite it in a standard quadratic form:

\[ A(w) = w^2 + 5w \]

This is a quadratic function that opens upwards (since the coefficient of \( w^2 \) is positive). The vertex of a quadratic function in the form \( f(w) = aw^2 + bw + c \) occurs at \( w = -\frac{b}{2a} \).

For our function:

• \( a = 1 \)
• \( b = 5 \)

Thus, the vertex \( w \) is computed as:

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

Since the function opens upwards, this means the minimum value occurs at \( w = -\frac{5}{2} \) and the function has no maximum value. Therefore, the correct relationship that describes \( A(w) \) is:

A(w) increases as w increases for all w > -2.5.

Now, looking at the original options presented:

1. A(w) has a minimum value at w=5.
2. A(w) increases as w increases when w > 0.
3. A(w) increases as w increases for 0 < w < 5.
4. A(w) has a maximum value at w=5.

The most accurate response based on the characteristics of the quadratic function is:

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

This is because while \( A(w) \) is a quadratic function with minimum value at \( w = -2.5 \), we can say \( A(w) \) does indeed increase for \( w > 0 \). Hence, this is the best option among the responses provided.