Question

Now that the satellite has been launched (i.e. Ch. 12 Lab), Earth is hopeful to get a more reliable eye on space anomalies. Shortly after the satellite's course around Jupiter, communication started to get more and more sporadic and weaker in signal. To counter this, a new satellite dish will be designed, specifically targeting the new satellite. Satellite dishes follow the shape of a paraboloid. The equation of a paraboloid is as follows: z=x^2/a^2+y^2/b^2 Where a and b could be the same number (depending on the desired shape). When you are designing this dish, where should you place the receiver? The receiver is like the ‘antenna’, receiving the data signals. Could the effectiveness of the receiver be changed if the ratio a/b is changed? A preliminary design is started with the dish’s circular radius of 20 feet, and a depth of 5 feet. What is the equation for this paraboloid? Hint: a = b, for this design. When the dish is a circular paraboloid (again a = b), a two-dimensional model could be viewed with the equation: x^2=4py , where p is the distance from the parabola's vertex to its focus. This equation can be revolved around the y-axis to form the shape of the satellite dish. Using the dimensions of the dish in #2, find the equation of the parabola in two-dimensions. Where should the receiver be placed? If the graph is revolved around the y-axis, what is the surface area of the satellite dish? From #3, find the cost of the dish itself when the material to make the dish costs $1,500 per square foot (it is a high quality satellite dish!). Many satellite dishes today are elliptic paraboloids (now a ≠ b), for reasons of reducing interference. But, would this change of shape also change where to place the receiver? Let’s use a satellite dish which has a 20-foot radius in the x-direction, and has a 10-foot radius in the y-direction. If a single receiver is placed at 10 feet from the vertex, what must the depth be in the x-direction and y-direction, separately, in order that the signal is maximized? What would a function look like for the depth of the dish at any particular place on the rim? Is there any theoretical reason to use an elliptic paraboloid vs. a circular paraboloid? Explain your reasoning. Thank you.

Answer 1

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