Question

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 sho calculator

Explanation:

Step1: Expand the area function

First, we expand the given area function \( A(x)=-x(x - 80) \). Using the distributive property (also known as the FOIL method for binomials), we get:
\( A(x)=-x^{2}+80x \)
This is a quadratic function in the form \( y = ax^{2}+bx + c \), where \( a=- 1 \), \( b = 80 \), and \( c = 0 \).

Step2: Find the vertex of the parabola

For a quadratic function \( y=ax^{2}+bx + c \), the \( x \)-coordinate of the vertex (which gives the maximum or minimum value of the function, depending on the sign of \( a \)) is given by the formula \( x=-\frac{b}{2a} \).
Since \( a=-1<0 \), the parabola opens downwards, and the vertex represents the maximum point of the function.
Substitute \( a=-1 \) and \( b = 80 \) into the formula:
\( x=-\frac{80}{2\times(-1)} \)
First, calculate the denominator: \( 2\times(-1)=-2 \)
Then, \( x =-\frac{80}{-2}=40 \)

Answer:

40