Find the standard form of the equation of the parabola with the given characteristics. vertex: (5, 25) points on the paraboia: (0, 0), (10, O) y : ix? + 10x A solar collector for heating water is constructed with a sheet of stainless steel that is formed into the shape of a parabola with width w = 18 meters (see figure). The water will flow through a pipe that is located at the focus of the parabola. At what distance from the vertex is the pipe? xm LORAN (long distance radio navigation) for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light (185,000 miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola having the transmitting stations as foci. Assume that two stations, 300 miles apart, are positioned on a rectangular coordinate system at (7150, 0) and (150, O) and that a ship is traveling on a path with coordinates (x, 75) (see figure). Find the xicoordinate of the position of the ship if the time difference between the pulses from the transmitting stations is 500 microseconds (0.0005 second). (Round your answer to one decimal place.) Use a graphing utility to graph the curve represented by the parametric equations. (Indicate the orientation of the curve.) x = 6 sin(28), y = 7 605(28) B Eliminate the parameter and write the corresponding rectangular equation. X Ellminate the parameter and obtain the standard form of the rectangular equation. Line through (x1, yl) and (x2, 5'2): X=X1+5(X2X1), Y=Y1+Y2y1) Use your result to find a set of parametric equations for the line or conic. (Enter your answers as a comma-separated list.) Line: passes through (0, 0) and (4, 2) X Eliminate the parameter and obtain the standard form of the rectangular equation. Circle: X : h + r cos(6'), y : k + r sin(9) Use your result to find a set of parametric equations for the line or conic. (When 0 S 9 S 211: Enter your answers as a comma? separated list of equations.) Circle: center: (3, 3); radius: 2 X \fFind dy/dx and day/dx , and find the slope and concavity (if possible) at the given value of the parameter. (If an answer does not exist, enter DNE.) Parametric Equations Point x = vt, y = 6t - 1 t = 9 dy DNE dx X day 12 dx 2 slope DNE XFind all points (if any) of horizontal and vertical tangency to the curve. Use a graphing utility to confirm your results. (If an answer does not exist, enter DNE.) x = 3 - t, y =-+2 Horizontal tangent ( x, y ) = 3 - t, -72 X