To determine which quadratic equation correctly represents the area [tex]\(A\)[/tex] of a rectangle in square feet given its width [tex]\(w\)[/tex] in feet, let's analyze the options provided.
Let's denote the length of the rectangle by [tex]\(l\)[/tex] feet and the width by [tex]\(w\)[/tex] feet. The area [tex]\(A\)[/tex] of the rectangle can be expressed as:
[tex]\[ A = l \cdot w \][/tex]
Since we have a quadratic equation related to the area:
[tex]\[ A(w) = aw^2 + bw + c \][/tex]
we need to identify which form aligns best with the general characteristics of a quadratic equation for area.
Given the choices:
1. [tex]\( A(w) = -w^2 + 200w \)[/tex]
2. [tex]\( A(w) = -w^2 + 100w \)[/tex]
3. [tex]\( A(w) = w^2 + 40w \)[/tex]
4. [tex]\( A(w) = w^2 + 90w \)[/tex]
We should notice that in a physically realistic context, the area of the rectangle will generally increase initially with width [tex]\(w\)[/tex] but eventually might see a maximum value and decrease if constrained by some higher relationship. This is often reflected in the first term being negative ([tex]\(-w^2\)[/tex]), indicating a parabola that opens downwards.
Examining all given options, the correct quadratic equation providing the area [tex]\(A\)[/tex] of the rectangle in square feet given its width in [tex]\(w\)[/tex] feet, while acknowledging the overall relationship involving quadratics, matches:
[tex]\[ A(w) = -w^2 + 200w \][/tex]
Therefore, the correct quadratic equation that gives the area [tex]\(A\)[/tex] of the rectangle in square feet given its width in [tex]\(w\)[/tex] feet is:
[tex]\[ A(w) = -w^2 + 200w \][/tex]
