Algebra II
Name: ___________________________________ Period: _________ Unit 5: Conic Sections Real World Applications Project
Part 1 (25 points): You will create a collage of pictures illustrating all four conic sections (circles, ellipses, parabolas, and hyperbolas) found in nature (leaves, flowers, body parts, etc.), architecture (bridges, doorways, etc.), and everyday items (appliances, logos, furniture, etc.). Requirements: Your project must contain: 1. Pictures of the entire objects where the conic section is found 2. Five different examples for each of the four conic sections: circles, ellipses, parabolas, and hyperbolas (no repeat pictures are allowed) 3. Trace, in marker, the conic in each picture 4. Title for the poster 5. CREATIVITY!!! Illustration: Your collage should be created using the following restrictions: * white or colored poster board * use scissors to cut out pictures (no tearing) * use glue to paste pictures (no taping)
Part 2 (25 points): Congratulations! You have just been upgraded from the status of Metro student to Metro Conics Artist! The job has been given to you to help critique your mastery and ability of graphing and creating equations of conic sections. You will be creating a MASTERPIECE, a work of art, that proudly displays conic sections. But of course, you will also need to display their equations, as well. You must design an image that represents at least one of the real world objects (nature, architecture, or everyday item) you found in Part 1. Your art work must contain: - At least 10 conic sections (2 of each conic – parabolas, circles, ellipses, and hyperbolas) - Equations for ALL conics used in your art work neatly labeled and written - At least 4 domain restrictions (your conic will not necessarily go on forever and forever and you will need to show where you are stopping them on your equation sheet) - At least 5 conics NOT CENTERED at the origin - COLOR!! - On the same paper as your equations, I will also like for you to answer the following questions: 1) What did you learn from doing this project? (At least 5 sentences) 2) What difficulties did you face and how did you solve them? (At least 10 sentences) 3) How did you come up with the idea of the picture? (At least 5 sentences)
Part 3 (25 points): You will answer the following questions on a separate sheet of paper. Clearly label the questions and show all necessary steps. 1. A satellite dish is shaped like a paraboloid of revolution. The signals that emanate from a satellite strike the surface of the dish and are reflected to a single point, where the receiver is located. If the dish is 8 feet across at its’ opening and 3 feet deep at its’ center, at what position should the receiver be placed?
2. The towers of a suspension bridge are 400 feet apart and 100 feet high. Cables are at a height of 10 feet between the towers. Assume the x-axis is the road and the y-axis is the center of the bridge, write an equation for the parabola. What is the height of the cable at a point 50 feet from one of the towers?
3. The parabolic arch shown in the figure is 50 feet above the water at the center and 200 feet wide at the base. Will a boat that is 30 feet tall clear the arch 30 feet from the center?
4. Two towers OP and RQ of a suspension bridge are 50 meters high and 60 meters apart. A cable is suspended between P and Q and approximates the shape of a parabola. The maximum sag in the middle of the cable is 20 m. 1. Find the coordinates of the vertex of the parabola. 2. Find the equation of the parabola. 3. How high is the cable directly above X, given that XR = 10 meters?
5. Use the following diagram of the bridge to answer each question. Assume the road is the x-axis and the center of the bridge is the y-axis. 1. Write an equation of the parabolic support. 2. Find the height of the vertical support 15 meters from the center.
6. The design of a garden is shown at the left. A pond is to be built in the center region. What is the equation of the largest circular pond centered at the origin that would fit within the walkways?
7. The University of Southern California is located about 2.5 miles West and about 2.8 miles South of downtown L.A. Suppose an earthquake occurs with its’ epicenter about 40 miles from the university. Assume that the origin of a coordinate plane is located at the center of downtown L.A. Write an equation for the set of points that could be the epicenter of the earthquake.
8. The Whispering Gallery in the Museum of Science and Industry in Chicago is 47.3 feet long. The distance from the center of the room to the foci is 20.3 feet. Find an equation that describes the shape of the room. How high is the room at its’ center?
9. A racetrack is in the shape of an ellipse, 100 feet long and 50 feet wide. What is the width of the racetrack 10 feet from the vertex?
10. An arch in the shape of the upper half of an ellipse is used to support a bridge that is to span a river 20 meters wide. The center of the arch is 6 meters above the center of the river. Write an equation for the ellipse in which the x-axis coincides with the water level and the y-axis passes through the center of the arch.
11. A satellite is in elliptical orbit around the earth with the center of the Earth at one focus. The distance of the satellite from the Earth varies between 140 mi and 440 mi. Assume the Earth is a sphere with radius 3960 miles. Find an equation for the path of the satellite.
12. A forester at an outpost and another forester at the primary station both heard an explosion. The outpost and the primary station are 6 kilometers apart. If one forester heard the explosion 6 seconds before the other, write an equation that describes all the possible locations of the explosion. Place the two forester stations on the x-axis with the midpoint between the stations at the origin. The transverse axis is horizontal. (Hint: The speed of sound is about .35 kilometer per second). Draw a sketch of the possible locations of the explosion. Include the ranger stations in the drawing.
Grading: You will be graded according to the following rubric:
Part 1: Category
Points Possible
5 examples of circles
5 points (1pt for each)
5 examples of ellipses
5 points (1pt for each)
5 examples of parabolas
5 points (1pt for each)
5 examples of hyperbolas
5 points (1pt for each)
Examples of nature (at least 5)
5 points (1pt for each)
Examples of architecture (at least 5)
5 points (1pt for each)
Examples of everyday items
10 points (1pt for each)
Tracing of the conic sections
20 points (1pt for each)
Title
2 points
Neat/Unique/Appropriate
Up to 8 extra points
Points
Total Points: 70 Scaled Down to Being out of 25 points: ________ / 25
Part 2: Points Technical Correctness
Variety of Conics Used
Difficulty Level
1 There are more than five errors in the equations used to represent the curves of the artistic rendering or the equations were not given. There are seven or more conics not in the technical drawing
Conics that are included do not have the appropriate domain restrictions.
3 The equations representing curves in the artistic rendering have three to five errors.
4 The equations representing curves in the artistic rendering have one or two errors.
5 The equations representing curves in the artistic rendering are correct.
One to three of the required ten conics are not represented in the technical drawing. All conics are included, but no domain restrictions are used.
Ten conics are represented in the technical drawing.
At least three additional conics are included over the required ten conics.
The conic sections represent the appropriate parabola, ellipse, circle and
The conic sections represent parabola, ellipse, circle and hyperbola are not centered at the origin.
Organization
Artistic Rendering
The project is not organized and cannot be followed.
The project is not separated into organized parts: equations and graphs. It is hard to follow along.
The artistic rendering has four or more errors in its representation of the technical drawing.
The artistic rendering does not represent the technical drawing in one to three ways or is not colored.
hyperbola with at least three domain restrictions as appropriate. The project is presented with the artistic rendering, the technical drawing, the equations are easy to follow. The artistic rendering represents the technical drawing and color is added.
Conics included are those with at least four domain restrictions with a variety of orientation. The project is presented in a professional format. All equations are typed into a word document or neatly written.
The artistic rendering represents the technical drawing and contains extra details that make it attractive and a complete drawing.
TOTAL POINTS: __________ / 25
Part 3: Category
Points Possible
Accuracy
12 (1 per question)
Work Shown
12 (1 per question
Organization
1
Points
TOTAL POINTS: __________ / 25
