91Ó°ÊÓ

Quadratic functions are a fundamental concept in calculus and algebra. They form the basis for understanding many real-world phenomena. A quadratic function can be expressed in the general form:

• \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \).

The graph of a quadratic function is a parabola, which can be either "smiling" (opens upwards) or "frowning" (opens downwards) depending on the sign of the coefficient \( a \). A downward-opening parabola indicates that the coefficient of \( x^2 \) is negative. In the context of this exercise, the function \( A(w) = -w^2 + 21w \) implies the parabola opens downwards, which is why it has a maximum point rather than a minimum.
This maximum point can be found using the vertex formula \( x = -\frac{b}{2a} \). Substituting the values from our function: \( a = -1 \) and \( b = 21 \), we find the \( w \) that maximizes the area is \( 10.5 \). At this point, the area is maximized thanks to the properties of quadratic functions.

The area of a rectangle is one of the simplest, yet most important, geometric concepts. It helps us understand the size of two-dimensional spaces. The area \( A \) is calculated as the product of its length \( l \) and width \( w \):

• \( A = l \times w \)

In optimization problems, such as the one we are solving here, our goal is to maximize this area under certain constraints, like the perimeter. In this scenario, we formulated the area equation in terms of one variable by expressing \( l \) through \( w \) as \( l = 21 - w \), based on the perimeter condition \( 2l + 2w = 42 \).
By substituting \( l \) in the area formula, we calculate the function \( A(w) = (21 - w)w = 21w - w^2 \). This way, the problem of maximizing area translates to finding the vertex of the quadratic function representing area, a task perfect for calculus techniques.

Perimeter is the total length of the boundaries of a shape. For rectangles, the perimeter \( P \) is calculated by adding up all the sides and is given by the formula:

• \( P = 2l + 2w \)

This formula is essential when dealing with optimization problems, as it often serves as a constraint for other calculations, like maximizing area. In our exercise, the perimeter constraint of \( 42 \) feet allowed us to express one variable in terms of another (\( l \) in terms of \( w \)), which is a crucial step in simplifying and solving the problem.
Knowing the perimeter enables the use of algebra to find relationships between dimensions, ensuring solutions are precise and efficient. By dividing the perimeter equation \( 2l + 2w = 42 \) by 2, the simplified equation \( l + w = 21 \) creates a direct relationship between length and width, facilitating the substitution into other equations to solve for optimal values.