Algebra 2 end of year project

Name _________________________________ I.D. Number ________________________ Project 2 Evaluation 32 Second Year Algebra 2 (MTHH 040 059) Part B – Applications of Parabolas & Ellipses (possible 30 points) Activity: You will answer the following questions. Clearly label the questions and show ALL necessary steps. 1. A satellite dish is shaped like a parabola. The signals that originate 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 16 feet across at its opening and 5 feet deep at its center, at what position should the receiver be placed? (2 pts) ! 2. A searchlight is shaped like a parabola. If the light source is located 3 feet from the base along the axis of symmetry and the depth of the searchlight is 4 feet, what should the width of the opening of the searchlight be? (3 pts) 3. The towers of a suspension bridge are 450 feet apart and 150 feet high from the roadway. Cables are at a height of 25 feet above the roadway, midway between the towers, but gradually get taller toward each end. Assume the x-axis is the roadway 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? Round to the nearest whole number. (3 pts) Project 2 MTHH 040 4. The parabolic arch shown in the figure is 45 feet above the water at the center and 180 feet wide at the base. Will a boat that is 30 feet tall clear the arch 30 feet from the center? (Use water as the x-axis.) (3 pts) ! 5. The Whispering Gallery in the Museum of Science and Industry in Chicago is 47.3 feet long and is in the shape of an ellipse. 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? (3 pts) 6. 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? Make sure to identify both endpoints of the new width. (3 pts) 7. A bridge is to be built in the shape of a parabola and is to have a span of 140 feet. The height of the arch at a distance of 50 feet from the center is to be 20 feet. Find the height of the arch at its center. (3 pts) Project 2 MTHH 040 Part C – Martian Math (possible 12 points) Activity: Even life has not yet been discovered on Mars, today we know a lot of other things about the planet. Mars has a day about 25 hours long, a pattern of seasons similar to Earth’s, and polar icecaps. Mars also has surface temperatures that rarely rise above freezing and almost no oxygen in its atmosphere. Mars is often called the Red Planet because red deserts cover its surface. ! 1. Mars travels in an elliptical orbit with the Sun at one of its foci. Use the data from the diagram to calculate a, b, and c of this elliptical orbit. (4 pts) 2. Point P is the midpoint between Mars’ closest and farthest distances to the Sun. Use your values of a and b to write an equation of the elliptical orbit of Mars relative to a coordinate system drawn through point P (y-axis in gray). Use distances in millions of kilometers. (3 pts) ! 3. It is also useful to define Mars’ motion relative to the Sun. Imagine a new coordinate system (yaxis in black) with its origin at the center of the Sun. Rewrite your equation for the ellipse in this new coordinate system.