Problem 60 - College Algebra with Corequisite Support

Question

The height \(h\) of a projectile is a function of the time \(t\) it is in the air. The height in feet for \(t\) seconds is given by the function \(h(t)=-16 t^{2}+96 t .\) What is the domain of the function? What does the domain mean in the context of the problem?

Step-by-Step Solution

Answer

The domain is \( 0 \leq t \leq 6 \). It represents the time (in seconds) the projectile is in motion.

Step 1: Identify the Function's Components

The function given is \( h(t) = -16t^2 + 96t \). It's a quadratic equation representing the height of a projectile over time. The coefficients \(-16\) and \(96\) determine the parabola's orientation and intercepts.

Step 2: Determine the Time of Flight

Since the projectile's path starts and ends on the ground (height = 0), we need to find when \( h(t) = 0 \). Setting the equation \(-16t^2 + 96t = 0\), factor out \( 16t \), resulting in \( 16t(-t + 6) = 0 \). This gives the zeros \( t = 0 \) and \( t = 6 \).

Step 3: Establish the Domain in Context

In this real-world scenario, \( t \) (the time the projectile is in the air) must be between \( 0 \) and \( 6 \) seconds inclusively, as these are the times it leaves and returns to ground level. Thus, the domain is \( 0 \leq t \leq 6 \).

Step 4: Interpret the Domain

The domain \( 0 \leq t \leq 6 \) means the function is valid from the time the projectile is launched to when it hits the ground again. The function does not represent physical reality for \( t < 0 \) or \( t > 6 \), as there is no projectile movement at those times.

Key Concepts

Quadratic Function

A quadratic function is a polynomial function of degree 2. Its standard form is expressed as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( x \) represents the variable. In our specific context, the quadratic nature of the function defines the path of a projectile moving under the influence of gravity. Quadratic functions create a parabolic shape on a graph, which is why they are perfect for modeling projectile motion.

Let’s break this down:

• **Coefficient \( a \)**: This determines whether the parabola opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)). Here, \( a = -16 \) suggests a downward-opening parabola.
• **Vertex**: The peak or the lowest point of the parabola, offering crucial information about the projectile’s maximum height or minimum in other cases.
• **Intercepts**: These are where the graph crosses the axes, providing launch (initial height) and landing points (final height).

Understanding the components of a quadratic function helps in interpreting how changes in time affect the trajectory and height of the projectile.

Domain of a Function

The domain of a function is a set of all possible input values (or \( x \)-values) that the function can accept and still yield a valid output. In mathematical terms, it defines the interval over which a function is defined and determines what inputs are "allowed" for practical calculations.

In our projectile problem, time \( t \) is the variable, and since it’s measuring the time the projectile is in the air, we need to find out when the height becomes impractical or undefined:

• **Initial and Final Moments**: Identify when the projectile is launched (\( t = 0 \)) and when it lands (\( t = 6 \)).
• **Real-World Limitations**: Time cannot be negative and a projectile doesn't hover after landing, setting realistic bounds on the function.

In this case, the domain is \( 0 \leq t \leq 6 \), which means that the function is only physically meaningful and applicable when time is between 0 and 6 seconds.

Parabolic Motion

Parabolic motion describes the path of an object that is projected into the air and subject to the force of gravity. It appears as a parabolic curve when graphed on a coordinate system.

This parabolic trajectory occurs due to the constant gravitational force pulling the object towards the Earth's surface while it moves forward horizontally:

• **Horizontal Motion**: Uniform and unchanging, as no horizontal forces are acting on the projectile (ignoring air resistance).
• **Vertical Motion**: Constantly accelerated, typically downward due to gravity, affecting the shape of the parabola.

When dealing with projectile motion, it's important to note how these factors—initial speed, angle, and gravitational force—combine to create the shape and characteristics of the projectile's path. This is where our quadratic function and its solution give us deeper insights into the practical aspects of motion under gravitational influence.