1023 10.3 Hyperbolas (a) Determine the equation of the trajectory of the alpha particle if d = 5 * 10-14 m. (b) What was the minimum distance between the centers of the alpha particle and the gold nucleus? Write the answer using scientific notation. Round to the nearest tenth. 64. LORAN System Ships and planes often use a location-finding system called LORAN. With this system, a radio transmitter at M in the figure sends out a series of pulses. When each pulse is received at transmitter S, it then sends out a pulse. A ship at P receives pulses from both M and S. A receiver on the ship measures the difference in the arrival times of the pulses. The navigator then consults a special map showing hyperbolas that correspond to the differences in arrival times (which give the distances d1 and d2 in the figure). In this way the ship can be located as lying on a branch of a particular hyperbola. x y y = x y = –x d A N x y 0 (–c, 0) (c, 0) P(x, y) The sound is detected at the closer microphone t seconds before being detected at the farther microphone. Assume that sound travels at a speed of 330 m per sec, and show that P must be on the following hyperbola. x2 3302t2 - y2 4c2 - 3302t2 = 1 4 M S P d1 d2 Suppose that in the figure, d1 = 80 mi, d2 = 30 mi, and the distance MS between the transmitters is 100 mi. Use the definition of a hyperbola to find an equation of the hyperbola on which the ship is located. 65. Sound Detection Microphones are placed at points 1-c, 02 and 1c, 02. An explosion occurs at point P1x, y2 having positive x-coordinate. See the figure.
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