Test Your Parabola Skills!Here is your chance to test your parabola knowledge and skills after our lesson on the conic. This is an application problem and is used in an attempt to bring real world situations to this area of mathematics. Luckily, the parabola is a widely used conic in our lives, so readily used in fact you might be surprised how many times you may have passed by one unknowingly. In this case, we will examine a satellite dish, shaped like a paraboloid of revolution.

The Arecibo Radio Telescope
The Arecibo Radio Telescope is the worlds largest radio telescope. Built in 1963, the Arecibo dish is located in sunny Puerto Rico and remains a wonder in structural engineering. It is operated by Cornell University for the National Science Foundation, though since 1992 it has been used extensively in SETI (The Search for Extraterrestrial Intelligence) investigations for radio signals from space. The absolutely astonishing size of the Arecibo dish and its ability to accurately locate signals allows it to detect even those of extreme faintness. It has a dish diameter of 305 meters, and is located 51.4 meters in a natural sinkhole beneath sea level. A key element to its correct calibration and operation is maintaining its almost 40,000 perforated aluminum panels, each measuring 1 meter by 2 meters. Let's get to the question! 
The Question!
Unfortunately, a recent hurricane has done damage to the famed telescope, destroying both the original focus, and an aluminum panel 50 meters from of the center of the structure. Engineers have decided their first concern is the focus, which allows the telescope to receive signals. First they must rebuild the focus, which means finding its exact position, and then they will need to know how far a technician will have to slide down the inside of the dish to reach the damaged panel. To the right is all of the information we know thus far. I also provided the equation for this type of parabola which opens along the yaxis. Once you have finished, proceed down the page for the answer. 
The Answer!
Finding the answer to this type of parabola application is extremely simple as you will soon see. First, to find the focus, "p", we must use the equation for this type of parabola, along with an available point for plug in variables of x and y. Since we know that one point is (152.5,0), lets plug these values in using y = 1/4p (x  h)^2 + k, the equation for a positive parabola with a yaxis of symmetry: 0 = 1/4p (152.5)^2  51.4. Now the rest is simple algebra as we solve for "p" as p = 113.11 meters from the vertex. Now that we know that 1/4p = 1/452.44, we have our completed equation, y = 1/452.44 (x  h)^2  51.4, we can find the how far down the technician must slide by plugging in 50 (you can pick positive or negative 50, because it is squared so it will not matter) for our x value, and solve for y. Our equation will look like this: y = 1/452.44 ( + 50)^2  51.4. You get a y value of 45.87. The technician must slide 45.87 feet down the inside of the satellite dish to reach the damaged panel and fix it. 