# Writing Functions

There are a lot of word problems which require an ability to correctly model a real-world problem with Algebraic functions. The following shows examples on writing and algebraic functions to model real-world situations.

The perimeter of a rectangular field is \( 80 meters \). Find the area of the field in terms of its length \( \mathbf{x}. \)

Solution:

Representations: P = Perimeter; A = Area

Let $P = 2x + 2y$ where $\mathbf{x}$ is the field's length and y is its width.

$A = xy$

Given: $P = 80$

$80 = 2x + 2y$

\(40 = x + y \) (divide both sides by 2)

\(40 - x = y \) (solve for y)

Solving for the Area of the given rectangular field, it follows that:

\(A = xy\)

\(A = x(40 - x)\) (substitute the derived value of y above)

\(A = 40x - x^2\)

\(A(x) = 40x - x^2 \)