Question

After traveling for 4 hours and 1800 miles, an airplane begins a steady descent. When it begins descending, its altitude (height) is 33,000 feet. After 8 minutes, it is at 17,000 feet.

Answer the questions to write a function that models the plane's altitude.

1. How quickly does the airplane descend? In other words, what is the rate of change of the plane's altitude? Show your work.

















2. When graphing the descent of the plane, what would the starting value for its altitude be?

















3. Write a linear function in y = mx + b form that gives the plane's altitude after it begins its descent. Use x to represent the time (in minutes) after the descent starts and y to represent the plane's altitude (in feet).

















4. Is there information given in this problem that you did not need to write an equation for the descent? If so, what unnecessary information were you given?

Answer 1

Answer:

Part 1)  Descending Rate = -2,000 ft/min

Part 2)  Starting altitude value is 33,000 ft

Part 3)  The linear equation for the plane's altitude (y) as a function of time (x) is:  $y=-2000\,\frac{ft}{min} \,x\,+\,33000\,ft$

Part 4)  The information not used for the equation is that the plane was travelling for 4 hours to cover 1800 miles before starting the descent

Step-by-step explanation:

Part 1)

The rate at which the airplane descends is given by the difference between final and initial altitude (17,000 ft-33,000 ft) divided the elapsed time (8 minutes):

$Descending\,\,Rate\,= \frac{17000-33000}{8} \,\,\frac{ft}{min}=\frac{-16000}{8} \,\,\frac{ft}{min}=-\,2000\,\,\frac{ft}{min}$

Part 2)

When graphing the descent (that is the plane's altitude as a function of time), the starting value for the altitude should be 33,000 ft

Part 3)

We can build the equation of the plane's altitude as a function of time in slope y-intercept form y = m x + b

by noticing that the slope "m" stands for the rate of descent that we found in part 1), and then using the information that at time zero (when the plane starts its descent), its altitude is 33000 ft:

$y=m\,x\,+\,b\\y=-2000\,\frac{ft}{min} \,x\,+\,b\\33000=-2000\,\frac{ft}{min} \,(0)\,+\,b\\33000=b$

with this information about the intercept "b", we can write the final expression for the plane's altitude as a function of time during its descent as:

$y=-2000\,\frac{ft}{min} \,x\,+\,33000\,ft$

Part 4)

The information that was not used to write the descent equation was the initial details about how long the plane traveled (4 hours) and for 1800 miles.