Problem 6 - Elementary Algebra

Question

Find the domain of each of the following equations. Assume that the independent variable is the variable that appears in the expression on the right side of the \("="\) sign. \(s=\frac{4}{9} t^{2},\) where this equation relates the distance an object falls, \(s,\) to the time, \(t,\) it has had to fall.

Step-by-Step Solution

Answer

Answer: The domain of the equation is \(t\geq0\) or \([0, +\infty)\).

Step 1: Analyze the given equation

The given equation is \(s=\frac{4}{9} t^{2}\), which is a quadratic equation with respect to time (\(t\)). It represents the distance an object falls (\(s\)) as a function of the time (\(t\)) it has had to fall. Since we are only interested in the time and distance where the object is falling, we need to find the domain of \(t\).

Step 2: Identify the domain restrictions based on the context

As we are dealing with time and distance in the context of a falling object, both quantities must be non-negative, i.e., \(t\geq0\) and \(s\geq0\). No other restrictions are imposed, so we are free to use any non-negative value of time (\(t\)) as input for this equation.

Step 3: Determine the domain of the given equation

Based on the information provided in step 2, the domain of the given equation is all non-negative values of \(t\), represented as \(t\geq0\). In interval notation, the domain of the equation can be written as \([0, +\infty)\). In conclusion, the domain of the equation \(s=\frac{4}{9} t^{2}\) is \(t\geq0\) or \([0, +\infty)\).

Key Concepts

Quadratic Equation

A quadratic equation is one of the basic types of polynomial equations and can be recognized by the presence of the squared term. The general form of a quadratic equation is:

• \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( x \) represents an unknown (often the independent variable).

In the context of the exercise, the equation \( s=\frac{4}{9} t^{2} \) is a simple form of a quadratic equation, where there is no linear term or constant term present; only the squared term \( t^2 \) with a coefficient. The equation relates to how the distance \( s \) varies with time \( t \) for a falling object in a proportional manner. Quadratic equations like this are often used to model situations in physics and other sciences where there is acceleration occurring.

Domain Restrictions

Understanding domain restrictions is essential when working with functions, as these restrictions determine the set of permissible input values for the independent variable. In mathematics, the domain of a function is the complete set of values for which the function is defined.

In real-world applications, like the fall of an object, additional context-based constraints are applied. Here, time can only be non-negative because we generally start measuring time from a point rather than "going back in time." Therefore, the domain restriction here is that \( t \geq 0 \). Similarly, we expect the distance to be non-negative because an object cannot have fallen a negative distance. This reinforces the domain restriction on time for this equation, confirmed by examining \( s = \frac{4}{9}t^2 \).

By analyzing the nature of the problem, it becomes apparent that the domain realistically consists of all non-negative values of \( t \), or mathematically represented as \([0, +\infty)\).

Independent Variable

The independent variable in a function is the variable whose variation does not depend on another. Instead, it represents the input or cause, while the dependent variable is the output or effect. In the equation \( s=\frac{4}{9} t^{2} \), time \( t \) is the independent variable.

The choice of independent variable influences how we interpret the function. Here, time influences the distance an object falls. This makes distance \( s \), the dependent variable, because it relies on the input values of time. As time progresses, distance increases accordingly. Therefore, \( t \) drives the function's behavior by defining input conditions that determine the outcome in practical applications. Understanding which is the independent variable is important because it allows us to map out the relationship of other variables in modeling real-world scenarios.