Related Rates II
Copyright Maplesoft, a division of Waterloo Maple Inc., 2007
Introduction
This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus� methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips. Click on the buttons to watch the videos.
The steps in the document can be repeated to solve similar problems.
Problem Statement
At 1:00 PM a ship traveling at 9 knots sets sail northeast along a line that makes a angle with a line running due east (see Figure 1). An hour later, a second ship sets sail due north. At 11 PM, the distance between the ships is observed to be 126 nautical miles and is increasing at a rate of knots. How fast is the northbound ship traveling?
Solution

Figure 1: Diagram showing the path of the ships.


Let be the displacement of the northbound ship and be the displacement of the ship traveling northeast, at time hours after 2:00 PM. Then the problem states:
At 11 PM:

Table 1: Figure 1 and the information given in the Problem Statement.

The distance between the ships is given by the law of cosines:

(3.1) 
To find the rate that the distance is increasing, differentiate this equation with respect to t, then evaluate the result at

To differentiate (3.1), rightclick on the equation and select Differentiate>t. Then right click on the new answer, select Evaluate at a point and choose t=9 as the point.



(3.2) 

(3.3) 
Substitute the known values from Table 1. In addition, let , the distance travelled by the northbound ship, and let , the speed of the northbound ship.

[Ctrl] drag a copy of the result above to a new line. Highlight each value to be replaced, and type in the new value. [Enter].



(3.4) 
This gives one equation in and . Find a second equation by evaluating the distance equation at .

Enter the eqation for distance by entering the equation label ([Ctrl][L]), then press [Enter]. Then right click , select Evaluate at a point and choose t=9 as the point.



(3.5) 

(3.6) 
Substitute the known values from the Table 1, including the definition for .

[Ctrl]drag a copy of the result above to a new line. Highlight each value to be replaced, type in the new value, and then press [Enter].



(3.7) 

(3.8) 

(3.9) 
Since both and need to be non negative, the solution for the northbound ship can be found above.
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