Example 3-8-13 - Maple Help

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Chapter 3: Applications of Differentiation



Section 3.8: Optimization



Example 3.8.13



Snell's law,

$\frac{\mathrm{sin}\left({\mathrm{θ}}_{1}\right)}{\mathrm{sin}\left({\mathrm{θ}}_{2}\right)}=\frac{{v}_{1}}{{v}_{2}}$

relates the speed of light in two adjacent media with the angles of incidence (${\mathrm{θ}}_{1}$) and refraction (${\mathrm{θ}}_{2}$). See Figure 3.8.13(a).

Derive Snell's law from Fermat's principle, namely, that light traverses the path for which travel time is least.

 > p1:=plottools:-arrow([0,3],[2,0],.05,.2,.1,color=red): p2 := plottools:-arrow([2,0],[3,-4],.05,.3,.1,color=green): p3:=plot([[2,3],[2,-4]],color=black): p4:=plots:-textplot({[1,.25,x],[2.5,-4.4,L - x]},font=[TIMES,ROMAN,10]): p5:=plot([[2,-4],[3,-4]],color=black,linestyle=2): p6:=plottools:-arrow([1.2,-5],[0,-5],.1,.3,.2,color=black): p7:=plottools:-arrow([1.8,-5],[3,-5],.1,.3,.2,color=black): p8:=plots:-textplot([1.5,-5,"L"],font=[TIMES,ITALIC,10]): p9:=plots:-textplot({[1.6,1.2,typeset(theta[1])],[2.2,-1.8,typeset(theta[2])]}): p10:=plots:-textplot({[.17,1.25,typeset(a)],[1.7,-2.5,typeset(b)]}): p11:=plots:-textplot({[.48,2.9,typeset(A)],[2.2,.2,typeset(C)],[3.3,-4,typeset(B)]}): plots:-display([p||(1..11)], scaling=constrained, tickmarks=[[4],[-6,0,4]],view=[0..3.5,default]);

 >

Figure 3.8.13(a)   Light passing from point $A$ to point $B$







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