91Ó°ÊÓ

To understand how to maximize the area of a rectangle with a given perimeter, we first need to know what the perimeter of a rectangle is. It's the total length around the rectangle and can be calculated using the formula:
\[ P = 2(l + w) \]
where \( l \) is the length and \( w \) is the width. This equation tells us that if you add the length and width together and then multiply that sum by two, you get the perimeter. For example, for a string of 14 inches, this whole length needs to cover the total perimeter of the rectangle.

• Given: \( P = 14 \) inches
• Equation: \( 2(l + w) = 14 \)

When solving perimeter-related problems, the idea is to express one of the dimensions (either \( l \) or \( w \)) in terms of the other. This simplifies the problem significantly when it comes to calculating other properties like area.

Quadratic equations are essential in figuring out how to maximize the area of our rectangle. A standard form of a quadratic equation looks like:
\[ ax^2 + bx + c = 0 \]
The equation developed from our rectangle problem is \[ A = 7l - l^2 \] which is a simplified form of a quadratic equation, given that we only need an area with respect to length. In this instance:

• \( a = -1 \) (coefficient of \( l^2 \))
• \( b = 7 \) (coefficient of \( l \))
• \( c = 0 \) (there is no constant term, or \( c \))

The shape of a parabola, as described by a quadratic equation with a negative \( a \), opens downwards. This is important, as it means there is a maximum point, characterized as the vertex. The vertex represents the peak, or in our case, the maximum area the rectangle can achieve.

The vertex of a parabola is a crucial element in this kind of optimization problem because it tells us where the maximum or minimum point is located. The vertex of a parabola given by the equation \( ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
For our area equation \( A = 7l - l^2 \), we use this vertex formula:

• \( a = -1 \)
• \( b = 7 \)

Plug these values into the formula to get:\[ l = -\frac{7}{2(-1)} = \frac{7}{2} = 3.5 \]— meaning the length that maximizes the area is 3.5 inches. With this length, since the perimeter equation expressed as \( l + w = 7 \), we find that the width is also 3.5 inches.
Thus, the maximum area is achieved when the rectangle forms a square with both dimensions equal to 3.5 inches.