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Chapter 3: Applications of Differentiation
Section 3.6: Related Rates
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Example 3.6.2
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At 1:00 PM a ship traveling at 9 knots sets sail north-east along a line that makes a angle with a line running due east. An hour later, a second ship sets sail due north, and at 11:00 PM, the distance between the ships is observed to be increasing at a rate of knots. Assuming it travels at constant speed, how fast is the north-bound ship traveling?
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Solution
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Mathematical Solution
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In the labeled diagram in Figure 3.6.2(a), is the distance sailed by the first ship; , the distance sailed by the second. The distance between the ships is .
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Let be the time at which the second ship starts.
This is 2:00, so at 11:00, this second ship will have sailed for 9 hours.
(The first ship will have sailed for 10 hours.)
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Since is not known, call this speed .
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At time , the first ship will have sailed nautical miles; the second ship, .
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See Table 3.6.2(a) for a summary of this data.
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Apply the law of cosines to the triangle shown in Figure 3.6.2(a), obtaining
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Set and solve for .
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Find
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>
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p1:=Student:-VectorCalculus:-PlotVector([<0,2>,<3*cos(Pi/6),3*sin(Pi/6)>],color=black,width=.1,head_length=.5):
p2:=plot([[3*cos(Pi/6),3*sin(Pi/6)],[0,2]],style=line,color=black):
p3:=plots:-textplot({[1.17,1.9,typeset(d(t))],[.19,1,typeset(b(t))],[1.1,.89,typeset(a(t))]},font=[Times,12]):
p4:=plots:-textplot({[.7,.14,typeset(Pi/6)],[.19,.37,typeset(Pi/3)]}):
plots:-display(p1,p2,p3,p4,scaling=constrained,tickmarks=[0,0]);
unassign('p1','p2','p3','p4'):
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Table 3.6.2(a) Data for the given problem
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Figure 3.6.2(a) Coordinate system for the ships
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Maple Solution
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Construct and obtain
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Construct the radical on the right-hand side of the law of cosines, replacing and with the expressions listed in Table 3.6.2(a).
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Context Panel:
Differentiate≻With Respect To≻
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Context Panel: Evaluate at a Point≻
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Solve for
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Using its equation label, set the derivative equal to and press the Enter key.
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Context Panel: Solve≻Solve
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Approximate the second solution
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Control-drag the second solution
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Context Panel: Approximate≻5
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The second solution is negative, and is therefore rejected. The first solution, namely, , gives the speed of the north-bound ship.
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