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Chapter 0: Problem 87

Sketching a Graph In Exercises \(87-90,\) sketch a possible graph of the situation. The speed of an airplane as a function of time during a 5 -hour flight

Short Answer

Expert verified
The resultant graph should begin and end at zero speed, showing an increase during the first hour, maintaining a relatively constant speed during the flight, and then decreasing before the end of the 5 hours as the plane begins to descend.

Step by step solution

01

Set Up The Axes

On a graph sheet, draw a horizontal line which represents the time in hours, and a vertical line which represents the speed. Mark the horizontal time axis from 0 to 5 in suitable parts representing each hour of the flight.
02

Initiate at Zero

At time \( t=0 \) , the flight is just starting, hence the speed is zero. Plot this point on the graph.
03

Ascend to Cruise

Within the first hour, the plane will reach its cruising speed which is its maximum speed. We can assume this happens around the 1-hour mark for the purpose of this graph. Indicate this point on the graph with a smooth curve rising from zero.
04

Cruising Phase

From around 1 hour to just before 5 hours, the plane maintains roughly the same speed (i.e., cruising speed) barring minor variations. This portion can be represented by a straight line for simplicity on the graph. Remember it will not be perfectly straight due to minor speed changes.
05

Descent

A little before 5 hours, the plane will begin descent and gradually decrease its speed. This can be represented by a smooth curve coming down from the cruising speed to zero at the 5-hour mark.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Functions of time
In calculus, understanding functions of time is crucial for depicting dynamic quantities that vary over specific periods. Time is an independent variable placed on the x-axis, and another variable, such as speed, is linked to it. In our scenario with the airplane, its speed is a function of time. This means that the speed depends on how much time has passed during the flight. Identifying this relationship is the first step in sketching a functional graph of any situation. For our graph, each point along the curve indicates a specific speed at a given time, from takeoff to landing.
Speed and time relationship
Understanding the speed and time relationship is key to interpreting flight dynamics. Here, speed is plotted against time to show how it changes during a 5-hour flight. Initially, the plane starts at zero speed as it prepares to take off. Over the first hour, the speed increases until it reaches a maximum cruising speed. This part of the graph translates to an ascending curve indicating acceleration.
  • The first phase shows acceleration as speed increases.
  • The plateau represents cruising, where speed remains steady.
  • The descent phase highlights deceleration as the plane reduces speed before landing.
By clearly understanding these phases, you can appreciate how speed relates to time.
Graph interpretation
Graph interpretation involves analyzing and understanding what a graph depicts about a situation. In graph sketching, such as with the airplane speed, it's not just about drawing lines; it's about understanding what those lines represent. Start by recognizing different sections of the graph:
  • Rising sections indicate increasing speed or acceleration.
  • Flat sections suggest steady speed or cruising.
  • Descending sections show reducing speed or deceleration.
By breaking down the graph into these parts, you interpret the different flight stages, providing insight into how the plane behaves over time. Each part of the graph tells a story about how speed varies with time, helping to visualize dynamic processes more effectively.
Airplane speed dynamics
Airplane speed dynamics refer to the changes in speed during different phases of a flight. These dynamics can be translated into graph segments. Initially, as the airplane takes off, its speed quickly increases, this is the climb to cruising altitude. The cruising phase represents a steady section, where the speed is mostly constant. Near the end of the flight, as the airplane descends, the speed decreases until it lands. This behavior can be effectively illustrated through a well-thought-out graph.
  • Takeoff: Speed increases rapidly until cruising altitude is reached.
  • Cruise: Speed is maintained at a steady level.
  • Descent: Speed decreases gradually as the airplane prepares to land.
Understanding these dynamics helps in accurately sketching a graph and appreciating how speed is managed throughout a flight.

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Most popular questions from this chapter

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