MA 15400 Lesson 11 Section 6.7Applied ProblemsThere are two methods of describing navigation. This first method is a method many students are familiar with; northwest, southeast, etc. In this method, the direction is always found from the 'north/south' axis, rather than from the x-axis that is usually used. A direction may be stated as N 40°W for example. This says from the north (positive part of y-axis), make an angle of 40° to the left (west). This method is used in certain navigation or surveying problems. The direction or bearing, from a point P to a point Q is specified by stating the acute angle that segment PQ makes with the north-south line through P. We also state whether Q is north or south and east or west of P. Notice that when this notation is used for bearings or directions, N or S always appears to the left of the angle and W or E to the right. These measurements will be of the form NθW, SθE, etc. The angle θ will always be an acute angle.Before we go into the application problems for this set, remember back to your days in Geometry? Remember when two parallel lines are cut by a transversal?1 and 1 are supplementary since they are the two angles that determine a line.1 = 2 and 1 = 2 since they are alternate interior angles1 and 2 are supplementary since they are same-side interior angles.1NSEW30357570PABCD2112Find the following bearings from the origin P to the given points.A: N 30° EB:C:D:MA 15400 Lesson 11 Section 6.7Applied ProblemsA ship leaves port(always consider port as the origin) and sails in the direction N78W at a rate of 35 mph. At the same time, another ship leaves the same port in the direction S12W at a rate of 21 mph.a) How far apart are the two ships after 3 hours? b) What is the bearing, to the nearest degree, from the first ship to the second?a) The Pythagorean Theorem can be used to find the distance between the two ships, since the distance is a hypotenuse.b) To find the bearing: Draw a new set of axes. To find the bearing, we need to know angle θ. However, we will have to use our right triangle to get the information to find θ. We can find angle α using a tangent function. We also must always use alternate interior angles. The angle formed by α and θ is an alternate interior angle to the 78° angle (using the N/S lines as the parallel lines).2SWhen the drawing is made, the line for the ship going NW is made longer because the rate is greater.NEW78°12°78 + 12 = 90° Since a straight angle is 180°, the angle adjacent to these angles must be 90°. A right triangle is formed.The distances for each ship for the 3 hours can bedetermined.63)3)(21(105)3)(35(21ddNEW78°12°θα7810563tan  Bearing: For all these types of problems to find bearing:Step 1: Find a pair of alternate interior anglesStep 2: Use a tangent functionStep 3: Use addition orsubtraction, depending on the picture of the problemdMA 15400 Lesson 11 Section 6.7Applied ProblemsIn air navigation, directions and headings are specified by measuring from the north in a clockwisedirection. In this case, a positive measure is assigned to the angle instead of the negative measure towhich we are accustomed for clockwise rotations. Measurements can be anywhere between 0 and 360 degrees; no N, S, E, or W are used. Due east would be 90° for example.3P.A30B45NNFind the headings from the origin P to the following points:A:B:MA 15400 Lesson 11 Section 6.7Applied ProblemsAn Airplane, flying at 300 miles per hour, flies in the direction 100 for 2 hours. It then flies in the direction 190 for 1.5 hoursa) Find the distance from the start point to the plane after the 3.5 hour flight.Is the triangle a right triangle?b) In what direction does the plane need to fly to get back to its start point? (We need to find θ in the picture below.)c) How long will it take for the plane to reach its start point?d = rt4NN100°ABCθα10°36010CFor all these types of problems to find bearing:Step 1: Find a pair of alternate interior anglesStep 2: Use a tangent functionStep 3: Use addition or subtraction, depending on the picture of the