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Chapter 3: Applications of Differentiation
Section 3.8: Optimization
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Example 3.8.8
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A hiker in a forest is at point , five miles from , the nearest point along a straight road bounding the forest. The hiker wishes to get to point , some eight miles down the road from . If the hiker can walk at 2 mph in the woods, and 4 mph along the road, where should the hiker emerge from the woods to complete the journey in least time?
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Solution
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Analysis
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As depicted in Figure 3.8.8(a), let be the distance from point that the hiker emerges from the woods. Then, is the distance the hiker walks along the road, and is the distance the hiker walks through the woods.
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Since rate × time = distance, the total time for the walk from to is
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Minimize subject to the implied constraint .
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>
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p1:=plot([[0,8],[0,0],[5,0],[0,2.5]],style=line,color=black):
p2:=plots:-textplot({[.4,1.25,typeset(x)],[1.1,5.75,typeset(8-x)],[2,.3,5],[3.5,2,typeset(sqrt(x^2+25))]},align=left,font=[default,12]):
p3:=plots:-textplot({[5,.4,typeset(A)],[.5,.3,typeset(B)],[.5,8,typeset(C)]},font=[default,14],align=left):
plots:-display(p1,p2,p3,axes=none,scaling=constrained);
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Figure 3.8.8(a) Schematic of hiker's path
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Graphical Solution
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The total travel time is graphed in Figure 3.8.8(b). There appears to be a minimum of approximately 4 between and .
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Of course, the graph can be probed to get a refined estimate of the minimum, but that is left to the reader.
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That the "dip" in graph of is so shallow is significant, suggesting that the travel time does not vary greatly with . In fact, the difference in travel time between the optimal, and the "least effort" one with , is small.
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Figure 3.8.8(b) Graph of
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Numeric Solution
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Control-drag the expression for the total time .
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Context Panel: Optimization≻Minimize (local)
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A minimal travel time of 4.17 hours is possible if the road is reached some 2.89 miles from point .
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Analytic Solution
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Define the objective function
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Control-drag
Context Panel: Assign Function
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Obtain the critical number
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Write and press the Enter key.
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Context Panel: Solve≻Solve
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Write the critical number.
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Context Panel: Approximate≻5 (digits)
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Apply the Second-Derivative test
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Context Panel: Simplify≻Simplify
(The extremum is a relative minimum.)
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Obtain the relative minimum time
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Evaluate at the critical number.
Press the Enter key.
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Context Panel: Simplify≻Simplify
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Context Panel: Approximate≻5 (digits)
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Check the times at the endpoint values of
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Evaluate at .
Context Panel: Evaluate and Display Inline
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Context Panel: Approximate≻5 (digits)
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Evaluate at .
Context Panel: Evaluate and Display Inline
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Context Panel: Approximate≻5 (digits)
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The relative minimum time is the absolute minimum time, hrs.
The most toilsome path () takes some 4.72 hours; but the optimal path, just under 4.17. The difference is about 33 minutes. The least toilsome path ( takes 4.5 hours, differing from the optimal path by just 20.1 minutes. The least toilsome path requires hiking through the woods for 5 miles; the optimal path, miles. Only the hiker knows if saving 20 minutes is worth hiking through an additional miles of woods.
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