Question

quadratic word problems (factored form)
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simon has 160 meters of fencing to build a rectangular garden.
the gardens area (in square meters) as a function of the gardens width x (in meters) is modeled by
a(x) = -x(x - 80)
what width will produce the maximum garden area?
meters
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Explanation:

Step1: Recall the property of quadratic in factored form

For a quadratic function in the form \( y = a(x - r_1)(x - r_2) \), the axis of symmetry (which gives the x - value of the vertex, and for a quadratic with \( a<0 \), the vertex is the maximum point) is given by the midpoint of the roots \( r_1 \) and \( r_2 \). The formula for the midpoint of two numbers \( r_1 \) and \( r_2 \) is \( x=\frac{r_1 + r_2}{2} \).

First, we find the roots of the quadratic function \( A(x)=-x(x - 80) \). To find the roots, we set \( A(x) = 0 \):
\( -x(x - 80)=0 \)
This implies either \( -x = 0 \) (so \( x = 0 \)) or \( x - 80=0 \) (so \( x = 80 \)). So the roots of the quadratic function are \( r_1 = 0 \) and \( r_2=80 \).

Step2: Calculate the x - value of the vertex (width for maximum area)

Using the formula for the midpoint of the roots (which is the x - coordinate of the vertex for a quadratic function in factored form), we have:
\( x=\frac{0 + 80}{2} \)
\( x=\frac{80}{2}=40 \)

We can also expand the quadratic function to the standard form \( y = ax^{2}+bx + c \) and use the formula \( x=-\frac{b}{2a} \) to verify.

Expanding \( A(x)=-x(x - 80) \):
\( A(x)=-x^{2}+80x \)
Here, \( a=-1 \) and \( b = 80 \). Using the formula \( x=-\frac{b}{2a} \):
\( x=-\frac{80}{2\times(-1)}=-\frac{80}{-2}=40 \)

Answer:

40