Problem 91 - College Algebra with Corequisite Support

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

(a) 1 second after launch, the height is 200 feet. (b) 2 seconds after launch, the height is 350 feet.

Step 1: Understanding the Function

The function given is \(h(t)\), which represents the height of the rocket above ground, in feet, with respect to time \(t\) in seconds. This means the function inputs a time and outputs the height of the rocket at that specific time.

Step 2: Analyzing Statement (a)

The statement \(h(1) = 200\) specifies that when \(t = 1\) second, the height of the rocket, \(h(t)\), is 200 feet above the ground. This implies that 1 second after launching the rocket, it has reached a height of 200 feet.

Step 3: Analyzing Statement (b)

The statement \(h(2) = 350\) specifies that when \(t = 2\) seconds, the height of the rocket, \(h(t)\), is 350 feet above the ground. This indicates that 2 seconds after launching, the rocket has ascended to 350 feet.

Key Concepts

Time-Height Relationship

In our exercise, we analyze the function \(h(t)\) which describes how the height of a rocket changes over time once launched. The time-height relationship is a fundamental concept in understanding rocket motion. In this context:

• The variable \(t\) represents the time that has passed since the rocket's launch.
• The variable \(h(t)\) provides us with the rocket's height at that specific time \(t\).

This function allows us to explore how rapidly or slowly the rocket gains altitude. For example, at \(t = 1\), the rocket reaches 200 feet, and at \(t = 2\) seconds, it climbs to 350 feet. This increase in height over time is crucial as it can help us determine the rocket's speed and acceleration.

Rocket Motion Analysis

Rocket motion analysis involves understanding how a rocket's height changes over different time intervals. By examining specific time points, such as \(h(1) = 200\) and \(h(2) = 350\), we can analyze the rocket's path.

• At \(t = 1\) second, the rocket has already reached a height of 200 feet, indicating its rapid initial ascent.
• At \(t = 2\) seconds, it reaches 350 feet, showing a continued increase in altitude.

This analysis helps in understanding the dynamics of the rocket's flight. We can determine how efficient and effective the rocket's propulsion is by observing these time-height data points.

• Speed is inferred from how the height changes between two time points.
• Evaluating different time intervals can show phases of acceleration or constancy in the upward motion.

Mathematical Modeling

Mathematical modeling involves using equations like \(h(t)\) to simulate and predict the behavior of a real-world object, such as a rocket. Modeling the time-height relationship is key in predicting future behavior based on known data.

• The equation \(h(t)\) organizes data into a form we can analyze and interpret.
• By solving the function at different values of \(t\), we discover points like \(h(1) = 200\) and \(h(2) = 350\), which offer insights into the rocket's performance.

Mathematical models are powerful for preparing and planning actual rocket launches. They allow scientists and engineers to predict trajectories, potential maximum altitudes, and the overall success of the mission.

• Understanding initial conditions helps in creating accurate models.
• Continued data analysis and refinement lead to improved simulations of the rocket's journey.

These models are invaluable tools in aerospace and many fields where predicting complex system behavior is necessary.