C 24 Match each description below with the most appropriate conic it describes below.
II. Applications of Hyperbolas
Example 3: Two signaling stations are 200 miles apart. A ship at sea receives a signal from one station 0.00038 seconds after receiving the signal from the other station. [Note: The speed of each radio signal is 186,000 miles per second.]
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HO #12 Answer Key to HO #11
Sound Off!
Real-World Applications of Conic Sections – Class Notes Filled In I. Applications of Parabolas
Example 1: A satellite dish is shaped like a paraboloid of revolution. The signals that emanate from the satellite strike the surface of the dish and are reflected to a single point, where the receiver is located. If the dish is 10 feet across at its opening and is 4 feet deep at its center, at what position should the receiver be located?
Solution: The receiver should be located at the focus.
Draw a picture, Place the vertex of the dish at (0, 0) (-5, 4) (5, 4) Label ordered pairs using the dimensions above
Find a by substituting (5, 4) into
yax
2 4a52 so… a =25
4 Use the relationship
c a
4 1
to find c the
distance from the focus to the vertex.
c 4 1 25 4 16 25 c
so the receiver should be placed at (0, 25/16) to be at the focus.
Example 2: The cables of a suspension bridge are in the shape of a parabola, as shown in the figure. The towers supporting the cable are 600 feet apart and 80 feet high. If the cables touch the surface of the road midway between the towers, what is the height of the cable at a point 150 feet from the center of the bridge?
Solution: You need to determine an ordered pair on the bridge with an x-coordinate of 150 ft.
Draw a picture using the information above. The bridge is 80 feet high and the towers are 600 ft. apart Find a by substituting (300, 80) into
yax
2 (-300, 80) (300, 80)2 300 80a so a = 1125 1 2 1125 1 x y and (150, y)
Find the height by substituting in 150 for x
2 ) 150 ( 1125 1
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II. Applications of Hyperbolas
Example 3: Two signaling stations are 200 miles apart. A ship at sea receives a signal from one station 0.00038 seconds after receiving the signal from the other station. [Note: The speed of each radio signal is 186,000 miles per second.]
Write the equation of the hyperbola representing the ships position. Solution: Draw a picture. Place each station on a foci 200 miles apart. The difference in time between the stations is equal to the major axis This means using the definition of a hyperbola we know
68 . 70 000 , 186 * 00038 . 0
2a miles this means a = 35.34
Since 2c = 200 then c = 100. Find b using a2b2c2
10000 9156
.
1248 b2
b28751.08
The equation of the hyperbola is 1
08 . 8751 92 . 1248 2 2 y x
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HO #13 Name:
Sound Off! Date:
Real-World Applications of Conic Sections – Class Activity Period:
Directions: Complete the following problems in groups. Draw a picture. Show all work. Real-World Problem Solving
Adapted from Precalculus by Michael Sullivan, 6th ed’n
1. A hall 100 feet in length is designed as a whispering gallery with an elliptical ceiling. If the foci are located 25 feet from the center, what is the height of the ceiling at its center? What is the equation of the ellipse used describe this gallery? You may assume it is centered at the origin.
2. An elliptical wall, 320 feet long and 150 feet wide is designed to be a whispering wall. How far would the listener have to be from the source of the sound in order to hear it?
3. A hall 64 feet in length is designed as a whispering gallery. If the ceiling reaches a height of 30 feet above the 5 foot vertical walls at the highest point, how far from the nearest walls should two people stand to be able to whisper to one another?
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4. The cables of a suspension bridge are in the shape of a parabola. The towers supporting the cable are 500 feet apart and 200 feet high. If the cables just touch the road surface midway between the towers, what is the height of the cable 75 feet from the tower?
5. A ship at sea receives a signal from one station 0.000625 seconds after it receives signal from another station. Write the equation of the hyperbola representing the ships possible location if the stations are 130 miles apart and the signals travel at 186,000 miles per second.
6. A racetrack is in the shape of an ellipse 80 feet long and 40 feet wide. What is the width 10 feet from the side?
80 ft 40 ft 10 ft ? ?
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HO #13 b Solutions Sound Off!
Real-World Applications of Conic Sections – Class Activity
Directions: Complete the following problems in groups. Draw a picture. Show all work. Real-World Problem Solving
Adapted from Precalculus by Michael Sullivan, 6th ed’n
1. A hall 100 feet in length is designed as a whispering gallery with an elliptical ceiling. If the foci are located 25 feet from the center, what is the height of the ceiling at its center? What is the equation of the ellipse used describe this gallery? You may assume it is centered at the origin.
and 502 – b2 = 252 so b = 43.3 Therefore, the height of the ceiling is 43.3 feet.
2. An elliptical wall, 320 feet long and 150 feet wide is designed to be a whispering wall. How far would the listener have to be from the source of the sound in order to hear it?
so, 1602-1502 = c2 c = 55.68 so the distance the listener must be from the source of the sound is 2 x 55.68 or 111.36 feet away.
3. A hall 64 feet in length is designed as a whispering gallery. If the ceiling reaches a height of 30 feet above the 5 foot vertical walls at the highest point, how far from the nearest walls should two people stand to be able to whisper to one another?
and 322 - 302 = 124 so the solution is 11.14 feet from the center of the room, or 32-11.14 = 20.86 feet from the wall.
4. The cables of a suspension bridge are in the shape of a parabola. The towers supporting the cable are 500 feet apart and 200 feet high. If the cables just touch the road surface midway between the towers, what is the height of the cable 75 feet from the tower?
Vertex is (0.0) and the parabola contains the points (-250, 200) and (250, 200).
The equation of the parabola is y = 0.0032x2 so when x = (250-75) = 175, then y = 98 feet high. The cable is 98 feet high at this point.
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HO #14 Name:
Sound Off! Date:
Research and Design Brainstorm
Period: Driving Question: Can you build a structure with a unique acoustic property?
Directions: To determine which structure interests your design group you should research examples and designs for structures with acoustic properties. After researching structures with acoustic properties your group should answer the questions below.
Here are some websites to support your research:
- http://en.wikipedia.org/wiki/Architectural_acoustics (Architectural acoustics)
- http://www.architechweb.com/ArticleDetails/tabid/254/ArticleID/3496/Default.aspx (More about architectural acoustics)
- http://en.wikipedia.org/wiki/Whispering_gallery (a collection of whispering walls and their locations)
Which structure interests your group?
Why does this structure interest your group?
What shape (conic section or 3-dimensional conic section) is used to create this structure?
What equations do you need to support your research?
What materials would you build your structure out of?
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HO #15 Name:
Sound Off! Date:
Design Rough Draft Worksheet
Period:
Directions: Complete each of the following. Teacher approval should be obtained after each step.
Sketch of Design with Measurements:
Equations with Analytical Support:
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HO #16 Name:
Sound Off! Date:
Peer Evaluations for Designs
Period:
Directions: Each group will complete an evaluation of each other group to help determine the final design of the structure.
1 – 5 Scale, 1 means no or unlikely while 5 means yes or very likely 1. Do you think it is feasible to build this structure?
1 2 3 4 5
Why?
2. Do you think this is an interesting example of a conic section with a unique acoustic property?
1 2 3 4 5
Why?
3. Do you think you would enjoy building and using this structure?
1 2 3 4 5
Why?
4. Is there an appropriate place for this structure in your selected community?
1 2 3 4 5
Why?
5. Do you think the materials and costs for this design are reasonable?
1 2 3 4 5
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HO #17 Name:
Sound Off! Date:
Group Summary Guidelines
Period:
Directions: Create a one-page summary containing your group’s responsibilities, your group’s final products and any mathematics or equations associated with your group’s tasks. You should also include information about how your group integrated into the entire group structure. You may use the space below for any notes.
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HO #18