Marquise has $200$ meters of fencing to build a rectangular garden. $\newline$ The garden's area (in square meters) as a function of the garden's width $\newline$ $x$ (in meters) is modeled by: $\newline$ $A(x)=-x^{2}+100x$ $\newline$ What side width will produce the maximum garden area? $\newline$ $\text{meters}$

Full solution

Q. Marquise has

200

meters of fencing to build a rectangular garden.

\newline

The garden's area (in square meters) as a function of the garden's width

\newline

x

(in meters) is modeled by:

\newline

A(x)=-x^{2}+100x

\newline

What side width will produce the maximum garden area?

\newline

\text{meters}

Identify Area Function: We are given the area function $A(x) = -x^2 + 100x$ , which represents the area of the garden in terms of its width $x$ . To find the width that produces the maximum area, we need to find the vertex of the parabola represented by the area function, since the vertex will give us the maximum value for a downward-opening parabola.
Find Vertex Formula: The area function $A(x) = -x^2 + 100x$ is a quadratic function in the form of $f(x) = ax^2 + bx + c$ , where $a = -1$ , $b = 100$ , and $c = 0$ . The $x$ -coordinate of the vertex of a parabola given by a quadratic function is found using the formula $-\frac{b}{2a}$ .
Calculate x-coordinate of Vertex: We apply the formula to find the x-coordinate of the vertex: $x = -\frac{b}{2a} = -\frac{100}{2*(-1)} = -\frac{100}{-2} = 50$ . This means that the width $x$ that will produce the maximum garden area is $50$ meters.
Verify Maximum Point: To ensure that we have found the maximum and not the minimum, we can check the coefficient of the $x^2$ term in the area function. Since the coefficient is $-1$ (a negative number), the parabola opens downwards, which means the vertex represents the maximum point.
Conclude Maximum Area: Now that we have found the width that gives the maximum area, we can conclude the solution. The side width that will produce the maximum garden area is $50$ meters.