Calculus, 8th Edition,stewart; page 20 problem 22

Question

An airplane takes off from an airport and lands an hour later at another airport, 400 miles away. if t represents the time in minutes since the plane has left the terminal building, let x(t) be the horizontal distance traveled and y(t) be the altitude of the plane.

(a) Sketch a possible graph of x(t).

(b) Sketch a possible graph of y(t).

(c) Sketch a possible graph of the ground speed.

(c) Sketch a possible graph of the vertical velocity.

Answer 1

Step:1

(a)

Graph :

Graph of the horizontal distance traveled by the plane :

The airplane travels a 400 miles in one hour.

When the plane reaches to a certain height, it will maintain a constant velocity.

There is a slight curve at the beginning and end of the plane because velocity changes.

axis : Distance travel by the plane.

axis : Time taken to reach the destination.

Step: 2

(b)

Graph :

Graph of the altitude of the plane :

There is a change in height at the beginning and ending of the plane.

In the rest of the time, the airplane maintains a same altitude thought out the journey.

axis : Altitude of the plane.

axis : Time taken to reach the destination.

Step: 3

(c)

Graph :

Graph of the ground speed :

Ground speed is the horizontal speed of an aircraft relative to the ground.

During take off of time the velocity increases, after that it maintains a constant velocity in the middle and while landing the plane the velocity decreases.

axis : Ground speed.

axis : Time taken to reach the destination.

Step: 4

(d)

Graph :

Graph of the vertical velocity :

Vertical velocity is the special type of velocity because velocity in the vertical direction is always affected by the acceleration due to gravity.

During take off the vertical velocity is positive and increases and it reaches to some height vertical velocity decreases to zero since the plane is not moving up or down.

axis : Vertical velocity.

axis : Time taken to reach the destination.

Solution:

(a) Graph of the horizontal distance traveled by the plane :

(b) Graph of the altitude of the plane :

(c) Graph of the ground speed :

(d) Graph of the vertical velocity :

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