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Using Hyperbolas to Locate Earthquakes Before we begin with this example, we need to review a couple of key facts about hyperbolas. Definition of a hyperbola: A hyperbola is the set of all points in a plane such that the difference of the distances from two fixed points is a constant. What this means is that if you have the foci of the hyperbola, F1 and F2, and point P on the hyperbola, PF2 – PF1 is always the same value, no matter where P is. (See the diagram to the right.) The question arose when we were creating the equation for hyperbolas, what is that constant value? We showed that PF2 – PF1 = 2a. Now we begin with the earthquake problem: Two earthquake-monitoring stations are 14 kilometers apart. At 3:45:17 am, Station 1 detects an earthquake that originated at an unknown point P. Station 2 detects the earthquake at 3:45:19 am, two seconds later. It is already known to the observers at both stations that the speed of the shockwave traveling through the ground is 5 km/sec. In addition, their clocks are synchronized, so they know that two seconds passed before the shockwave reached the second station after reaching the first. From this, they attempt to locate the epicenter of the earthquake (point P). How far did the shock wave travel during the two seconds between stations? Knowing that Distance = (Rate)*(Time), we calculate D = rt = (2 sec)(5 km/sec) = 10 km. This means that PS2 PS1 = 10 km. (Remember that PS2 = PS1 + 10. Therefore PS2 – PS1 = (PS1 + 10km) – PS1 = 10km.) This earthquake example fits the definition of a hyperbola perfectly. We have two fixed points (the two stations), and all that we know about point P is that the difference of its distances from those points is fixed (10 kilometers). This means that the two stations are the foci of a hyperbola, and the origin of the earthquake, point P, must be on that hyperbola. Now let’s try to find the equation of the hyperbola that the earthquake lies on: One of the facts about hyperbolas which we first showed was that for any point P on a hyperbola with foci F1 and F2, PF2 – PF1 = 2a. In this situation S1 and S2 are the foci and the earthquake is at point P. We just showed that PS2 PS1 = 10 km. We can conclude that 2a = 10 and a = 5. We also know that c = 7 (The foci are 14 kilometers apart, 2c = 14). Now we can find b2 to derive the hyperbola. We’ll let Station 1 be at the origin of our coordinate system. ⇔ ⇔ The hyperbola runs horizontally so the equation fits the form . We end up with the equation Since the earthquake’s shockwave reached Station 1 first, the epicenter must lie on the left side of the earthquake. But we still haven’t found the location of the earthquake. How is this done? With a third station collecting data, a second hyperbola can be created using station 1 and station 3 as the foci. The epicenter is at the intersection of these two hyperbolas. Locating the Epicenter of an Earthquake (50 points) Three stations record the times they detect an earthquake. They use these times to calculate an earthquake’s location. These stations are at points A, B and C. Station B is 24 kilometers east of station A. Station C is 12 kilometers north of station A. The shockwave travels through the ground at a speed of 5 km/sec. Let station A be at the origin of your coordinate system. At 2:00:00 pm, Station B detects the earthquake. C At 2:00:03 pm, Station C detects the earthquake. At 2:00:04 pm, Station A detects the earthquake. 12 km Useful formulas: Distance = speed × time A 24 km B All answers should be accurate to three decimal places. All questions answered in complete sentences. The first sheet should contain your work, equations and answers to questions. The second sheet should contain your Graphmatica graph. The third sheet should contain your full-size graph. 1) Let stations A and B be the foci of a hyperbola that the earthquake originated on. Write the equation of that hyperbola using the location of station A as the origin. 2) Let stations A and C be the foci of a hyperbola that the earthquake originated on. Write the equation of that hyperbola. 3) What are the coordinates of the earthquake? (Use the Tools > Find Intersection tool on Graphmatica to find the coordinates.) Choose a suitable grid range to show the stations and the epicenter (View > Grid Range). Print out your graph and hand it in with your other work. 4) On separate paper, choose a suitable scale and make a sketch of the two hyperbolas, the stations, and the location of the earthquake’s epicenter. Use a straight edge and draw a scale graph of the axis and the hyperbolas. Use the asymptotes to draw accurate graphs. Label the axis. (Use your and values to help draw your asymptotes.) 5) How far is the earthquake from station B? 6) How long did it take the earthquake’s shockwave to reach station B? 7) When did the earthquake occur (round to the nearest second)?
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