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FACULTY NOTES
The LTAs and Spinoffs are designed so that each professor can implement them in a way that is consistent with his/her teaching style and course objectives. This may range from using the materials as out-of-class projects with minimal in-class guidance to doing most of the work in class. The LTAs and Spinoffs are amenable to small group cooperative work and typically benefit from the use of some learning technology. Since the objective of the LTAs and Spinoffs is to support the specific academic goals you have set for your students, the Faculty Notes are not intended to be prescriptive. The purpose of the Faculty Notes is to provide information that assists you to take full advantage of the LTAs and Spinoffs. This includes suggestions for instruction as well as answers for the exercises.
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FACULTY NOTES
LTA 16
The Space Shuttle Landing
Learning Objective: The students will determine a piecewise function that models the final landing approach of the Space Shuttle.
Topics: Version I deals with setting up a coordinate system; deriving, graphing, and using linear functions; deriving, graphing, and using exponential functions; and basic right triangle trigonometry. Version II has two parts. Part A deals with setting up a coordinate system and deriving, graphing, and using linear functions. Part B involves deriving, graphing, and using piecewise functions.
Level: Version I is suitable for Precalculus students after they have studied exponential functions. If they are not familiar with piecewise functions, they can also do Part B of Version II. Version II is suitable for Intermediate Algebra, Technical Mathematics, and College Algebra students.
Mathematical Prerequisites: All students should be able to construct a suitable coordinate system and write the equation of a line. Students doing Version I should be able to use right triangle trigonometry and be familiar with exponential functions.
Technology: A graphing calculator is strongly recommended since students can store values and repeat calculations easily.
Suggested Strategies: Groups are strongly recommended for this LTA. You can have each group turn in one final report or you can have each group member write his or her own report, while using the group for problem-solving. We have found that students responded very well to this LTA when working in groups and were therefore able to complete much of the work outside of class. For both versions, it would be useful to read the introduction with the students in class before actually doing any of the work. This will set the stage and get the students thinking about the problem. Discuss the first part of the assignment – setting up the coordinate system and plotting the three data points. This is not as easy as it seems since the points cover large distances and yet all must be plotted so that they can be clearly seen. Encourage the groups to discuss their methods for constructing this graph, including pasting several sheets of graph paper together. We have had some students tape four sheets together, while others have preferred poster board. Have them come to the next class prepared to discuss any problems they had with the coordinate system. Intermediate Algebra, Technical Mathematics, and College Algebra students could do the beginning of Version II - modeling the first phase of the landing in class in their groups. The remaining two phases of the landing could be assigned for groups to do outside of class. Part B (piecewise functions) should probably be started with the groups in class; it could be completed by groups or individuals outside of class.
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Note on calculations: When a numerical answer requires more than one calculation, the full decimal place accuracy of the calculator should be used to obtain the answers given in the Solutions. Thus, if students round off intermediate answers and use them in subsequent calculations, their answers may differ somewhat from the ones given in the Solutions.
The data points are rising from left to right. That is, the farther the Shuttle is from the touchdown point, the higher it is above the ground. This is equivalent to saying that the nearer the Shuttle is to the touchdown point, the closer it is to the ground. Relative to the coordinate system, the Shuttle travels from right to left and moves down.
Slope =
The units of slope are feet of altitude per foot of directed distance from the runway threshold. The slope indicates how much the Shuttle descends for each foot that it nears the touchdown point. Possible answers: For every 32,100 ft (measured horizontally) that the Shuttle is closer to the runway, its altitude decreases by 11,615 ft. OR For every foot (measured horizontally) that the Shuttle is closer to the runway, its altitude decreases by about 0.36 ft.
- y = 0 36184_. x_ − 96378505_._
This function is a valid model for 7 , 500 ≤ x ≤ 39 , 600
Connect the points R(7500, 1750) and S(39600, 13365) in Exercise 1 with a line segment.
If x = 10000, y = 0.36184(10,000) − 963.78505 ≈ 2,655 ft
Glide Slope = 1
tan 19. 32,
− ^ ≈
o
The linear model for the transitional phase is: y = 0.33381 x − 753.
Glide Slope for a linear transitional function = 1
tan 18. 4,
− ^ ≈
o
- A linear function has constant slope, so if a linear model is used, the glide slope will not change during this phase. The line will not give a shallow glide slope as the Shuttle exits this phase.
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An approximate path shape for the transitional phase is shown below.
1,750 = ab^7500
131 = ab^2650
7500 1 4850 4850 2650
ab b b ab
a = ≈
(1.000534612)^2650
Thus, the exponential transition function is y = 31.78057(1.000534612) x The domain of the transitional model is 2,650 ≤ x ≤7,
The calculator answer is y = 31.78056592(1.000534612) x
Connect the data points Q(2650, 131) and R(7500, 1750) in Exercise 1 with the exponential function you obtained in Exercise 13. Students should evaluate the function at a few points between 2,650 and 7,500 and draw a smooth curve through them.
y = 0 02701. x +59 42268.. The domain of the final phase is − 2,200 ≤ x ≤2,
Connect the data points P(−2200, 0) and Q(2650, 131) in Exercise 1 with a straight line segment.
Glide Slope = 1
tan 1. 4,
− ^ ≈
o
- tan(
o) = ft 48 miles
d
⇒ d = 255 019, ≈
It would take the commercial aircraft 255,019 ft or about 48 miles. For the Shuttle, the total distance is 41,800 ft or about 8 miles. Thus, the commercial aircraft would take about 6 times as much distance to land.
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Distance from runway in ft x
Altitude in ft y
39,600 13, 7,500 1,
- Slope =
The units of slope are feet of altitude per foot of directed distance from the runway threshold. The slope indicates how much the Shuttle descends for each foot that it nears the touchdown point. Possible answers: For every 32,100 ft (measured horizontally) that the Shuttle is closer to the runway, its altitude decreases by 11,615 ft. OR For every foot (measured horizontally) that the Shuttle is closer to the runway, its altitude decreases by about 0.36 ft.
y = 0.36184 x − 963.78505 if x is between 7,500 and 39,600.
Connect the points R(7500,1750) and S(39600, 13365) in Exercise 1 with a line segment.
If x = 10,000, then y = 0.36184(10,000) − 963.78505 ≈ 2,655 ft.
Distance from runway in ft x
Altitude in ft y
7,500 1, 2,650 131
- Slope =
y = 0.33381 x − 753.60825 where x is between 2,650 and 7,500.
Connect the points Q(2650, 131) and R(7500, 1750) in Exercise 1 with a line segment.
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- A linear function has constant slope, so if a linear model is used, the slope will not change during this phase. The model will not describe a steep slope on entry and a shallow slope as the Shuttle exits this phase.
An approximate path shape that might accomplish the transitional phase is shown below.
Distance from runway in ft x
Altitude in ft y
2,650 131 −2,200 0
- Slope of the final linear phase =
y = 0 02701. x +59 42268. if x is between −2,200 and 2,650.
Connect the data points P(−2200, 0) and Q(2650, 131) in Exercise 1 with a straight line segment.
Part B
3,000 < x <12,
120 ≤ x ≤ 765
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- Writing the summary is more difficult for most students than doing all the mathematics in the exercises. If you are having the students hand in a group report, then discuss your expectations for this summary several days before it is due, and encourage the groups to discuss their summaries with you if they are unsure or are having difficulty deciding what to include. Stress that they should not discuss any of the actual calculations, but that they should clearly describe their methods of solution. The graph should look something like this (the extreme right hand portion of the graph has been omitted for the sake of clarity):
