Problem 91 - College Algebra

Question

Let \(h(t)\) be the height above ground, in feet, of a rocket \(t\) seconds after launching. Explain the meaning of each statement: a. \(h(1)=200\) b. \(h(2)=350\)

Step-by-Step Solution

Answer

At 1 second, the rocket is 200 feet high; at 2 seconds, it is 350 feet high.

Step 1: Interpret Statement a

The statement \( h(1) = 200 \) indicates that at \( t = 1 \) second after the rocket's launch, the rocket's height above the ground is 200 feet. The function \( h(t) \) represents height, and \( t \) represents time, so here, the rocket reaches a height of 200 feet shortly after launch, at precisely 1 second.

Step 2: Interpret Statement b

The statement \( h(2) = 350 \) suggests that 2 seconds after the rocket's launch, the height of the rocket is 350 feet. Just like in step 1, \( h(t) \) stands for the rocket's height at a given time \( t \), indicating that the rocket continues its climb and reaches 350 feet by 2 seconds.

Key Concepts

Height Function

In mathematics and physics, a height function is a way to represent an object's height above a reference point, usually the ground, at any given time. In our exercise, the height function is denoted by \( h(t) \), where \( t \) is the time in seconds after the rocket's launch. This function provides a precise measurement of how high the rocket is at each moment in time.

• The height function helps us to analyze and understand the motion of objects in a vertical direction.
• It provides important data on the position of the rocket as time progresses.
• This can be used to calculate the speed or acceleration of the rocket as well.

By interpreting values from the function, such as \( h(1) = 200 \) or \( h(2) = 350 \), we can easily determine the height of the rocket at specific times after launch.

Time Variable

The time variable \( t \) is a crucial component in functions like \( h(t) \), representing the number of seconds elapsed since an event started, in this case, since the rocket launch. This variable helps in tracking how changes occur over time.

• \( t \) begins at zero, representing the moment the rocket launches from the ground.
• By increasing \( t \), we get insights into how the rocket's height changes.
• The relationship between height and time is what forms the function \( h(t) \).

Understanding the time variable is key to interpreting the behavior of the function over time. It allows us to see how initial conditions and forces acting on the rocket influence its flight path.

Rocket Launch

A rocket launch is a complex event where a rocket propels itself from the ground to soar into the sky, often reaching considerable heights. This scenario provides a perfect real-life application for functions like the height function \( h(t) \).

• The launch involves powerful engines that generate thrust to overcome Earth's gravity.
• As the rocket ascends, its height at various seconds past launch is captured by the height function.
• Each second represents a snapshot of the rocket's journey skyward, showing gradual changes in altitude.

Studying a rocket launch through the lens of mathematics allows students to connect theoretical concepts to real-world activities, deepening their comprehension of physics and dynamics involved in launching a rocket.