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