Module 14  Related Rates  
Introduction  Lesson 1  Lesson 2  Lesson 3  Lesson 4  SelfTest  
Lesson 14.3: Two Ships  
A relatedrate problem that models two ships as they move away from each other is discussed in this lesson.
Two ships start at a point O and move away from that point along routes that make a 120° angle. Ship A moves at 14
Modeling the Positions of the Ships Suppose that Ship A is moving along the positive xaxis at 14 knots. Enter the parametric equations for Ship A's position at time t hours.
Parametric equations can be used to model the position of an object at time t that is moving at speed v along a line that forms an angle with the positive xaxis. Such equations have the form
x = vt cos Enter the equations for Ship B's position, which is moving at 21 knots along a line that forms a 120° angle with the path of Ship A.
Animating the Motion of the Ships
Make sure the Graph Order in the Graph Formats dialog box is still Simultaneous.
Finding the Distance between the Ships The distance c(t) between the ships can be found by using the distance formula: Use the restriction t 0 because the problem starts at t = 0.
Thus, . Finding the Speed at which the Ships are Moving Apart The speed at which the ships are moving apart can be found by finding the derivative of c(t) with respect to t, or just observing that it is knots from the formula for c(t). 14.3.1 Approximately how far apart are the ships after two hours? Click here for the answer. Modifying the Problem Assume that both ships travel in the same direction and at the same speed as before, but Ship A begins its journey 5 nautical miles from point O and Ship B begins 3 nautical miles from point O.
14.3.2 Find the derivative of c(t). Click here for the answer. 

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