An Airplane's Height In Feet During An Eight-hour Flight Is Given As $h(t$\]. Find The Expression That Represents The Rate Of Change Of The Plane's Height, With Respect To Time, Five Hours Into The Flight.A. $\frac{h(5)-h(0)}{5}$B.
Introduction
When an airplane is in flight, its height above the ground changes constantly. This change in height is a function of time, and understanding the rate of change of this height is crucial for pilots and air traffic controllers to ensure safe and efficient flight operations. In this article, we will explore the concept of the rate of change of an airplane's height and derive an expression that represents this rate of change.
The Concept of Rate of Change
The rate of change of a function is a measure of how quickly the function changes as its input changes. In the context of an airplane's height, the rate of change represents the speed at which the plane's height increases or decreases over time. Mathematically, the rate of change of a function f(x) is represented as the derivative of the function, denoted as f'(x).
Deriving the Expression for Rate of Change
Let's assume that the height of the airplane at time t is given by the function h(t). We want to find the expression that represents the rate of change of the plane's height, with respect to time, five hours into the flight. To do this, we need to find the derivative of the function h(t) with respect to time.
The derivative of a function f(x) is defined as:
f'(x) = lim(h → 0) [f(x + h) - f(x)]/h
In the context of the airplane's height, we can rewrite this definition as:
h'(t) = lim(h → 0) [h(t + h) - h(t)]/h
To find the rate of change of the plane's height at time t = 5, we need to evaluate the derivative h'(t) at t = 5.
Evaluating the Derivative at t = 5
To evaluate the derivative h'(t) at t = 5, we need to find the limit of the expression [h(5 + h) - h(5)]/h as h approaches 0.
However, we are not given the explicit function h(t). Instead, we are given the height of the airplane at time t = 5 and t = 0. We can use this information to find the rate of change of the plane's height at time t = 5.
Using the Given Information
We are given that the height of the airplane at time t = 5 is h(5), and the height of the airplane at time t = 0 is h(0). We can use this information to find the rate of change of the plane's height at time t = 5.
The rate of change of the plane's height at time t = 5 is given by the difference quotient:
[h(5) - h(0)]/5
This expression represents the average rate of change of the plane's height over the interval [0, 5].
Conclusion
In conclusion, the expression that represents the rate of change of the plane's height, with respect to time, five hours into the flight is:
[h(5) - h(0)]/5
This expression represents the average rate of change of the plane's height over the interval [0, 5]. We can use this expression to estimate the rate of change of the plane's height at time t = 5.
Answer
The correct answer is:
A.
Discussion
The rate of change of an airplane's height is a critical concept in aviation. Understanding the rate of change of the plane's height is essential for pilots and air traffic controllers to ensure safe and efficient flight operations.
In this article, we derived an expression that represents the rate of change of the plane's height, with respect to time, five hours into the flight. We used the given information about the height of the airplane at time t = 5 and t = 0 to find the rate of change of the plane's height at time t = 5.
The expression we derived is a measure of the average rate of change of the plane's height over the interval [0, 5]. We can use this expression to estimate the rate of change of the plane's height at time t = 5.
References
- [1] Calculus, 3rd edition, Michael Spivak
- [2] Calculus, 2nd edition, James Stewart
Note
Introduction
In our previous article, we explored the concept of the rate of change of an airplane's height and derived an expression that represents this rate of change. In this article, we will answer some frequently asked questions about the rate of change of an airplane's height.
Q: What is the rate of change of an airplane's height?
A: The rate of change of an airplane's height is a measure of how quickly the plane's height increases or decreases over time. It is a critical concept in aviation, as it helps pilots and air traffic controllers to ensure safe and efficient flight operations.
Q: How do you calculate the rate of change of an airplane's height?
A: To calculate the rate of change of an airplane's height, you need to find the derivative of the function h(t) with respect to time. The derivative of a function f(x) is defined as:
f'(x) = lim(h → 0) [f(x + h) - f(x)]/h
In the context of the airplane's height, we can rewrite this definition as:
h'(t) = lim(h → 0) [h(t + h) - h(t)]/h
Q: What is the difference between the rate of change and the instantaneous rate of change?
A: The rate of change is a measure of the average rate of change of the plane's height over a given interval. The instantaneous rate of change, on the other hand, is a measure of the rate of change of the plane's height at a specific point in time.
To find the instantaneous rate of change, you need to find the derivative of the function h(t) at a specific point in time. This is typically denoted as h'(t) = lim(h → 0) [h(t + h) - h(t)]/h.
Q: Why is the rate of change of an airplane's height important?
A: The rate of change of an airplane's height is important because it helps pilots and air traffic controllers to ensure safe and efficient flight operations. By understanding the rate of change of the plane's height, they can:
- Avoid collisions with other aircraft or obstacles
- Maintain a safe altitude and airspeed
- Plan efficient flight routes and altitudes
- Respond to changes in weather or air traffic conditions
Q: Can you provide an example of how to calculate the rate of change of an airplane's height?
A: Let's say we have an airplane that is flying at an altitude of 10,000 feet at time t = 0. At time t = 5, the airplane's altitude is 12,000 feet. We can use this information to calculate the rate of change of the airplane's height over the interval [0, 5].
The rate of change of the airplane's height is given by the difference quotient:
[h(5) - h(0)]/5
In this case, the rate of change of the airplane's height is:
(12,000 - 10,000)/5 = 2,000/5 = 400 feet per minute
Q: What are some common applications of the rate of change of an airplane's height?
A: The rate of change of an airplane's height has many applications in aviation, including:
- Flight planning and navigation
- Air traffic control and management
- Weather forecasting and prediction
- Aircraft performance and optimization
Conclusion
In conclusion, the rate of change of an airplane's height is a critical concept in aviation that helps pilots and air traffic controllers to ensure safe and efficient flight operations. By understanding the rate of change of the plane's height, they can avoid collisions, maintain a safe altitude and airspeed, plan efficient flight routes and altitudes, and respond to changes in weather or air traffic conditions.
References
- [1] Calculus, 3rd edition, Michael Spivak
- [2] Calculus, 2nd edition, James Stewart
- [3] Aviation Weather Services, Federal Aviation Administration (FAA)
Note
The rate of change of an airplane's height is a complex concept that requires a deep understanding of calculus and aviation principles. This article is intended to provide a general overview of the concept and its applications, rather than a detailed mathematical treatment.