Chapter 3: Applications of Differentiation
Section 3.8: Optimization
Example 3.8.6
Two posts are in a line perpendicular to a straight road, one at a distance of 20 m from the road, the other at a distance of 50 m. Where on the road is the angle formed by the lines of sight to the posts a maximum?
Solution
Analysis
Figure 3.8.6(a) is a representative sketch of the road (line OC), and the poles at A and B.
The observer is at point C, and the angle θ is to be maximized.
The distance between points C and O is denoted by x.
From right triangle OAC: tanα=20/x
From right triangle OBC: tanα+θ=50/x
Hence, θx=tan−150/x−tan−120/x
Figure 3.8.6(b) is a graph of θx.
module()
local f,P;
f := arctan(50/x)-arctan(20/x);
P:=plot(f,x=0..100,labels=[x,typeset(theta)]); print(P);
end module:
Figure 3.8.6(b) Graph of θx
p1:=plot([[0,0],[0,-7]],style=line,color=black): p2:=plot([[0,0],[5,0]],style=line,color=black): p3:=plot([[0,-7],[2,0]],style=line,color=red,linestyle=dot): p4:=plot([[0,-7],[5,0]],style=line,color=red,linestyle=dot): p5:=plot([[2,0],[5,0]],style=point,symbol=solidcircle,symbolsize=15,color=green): p6:=plots:-textplot({[-.2,-3.5,typeset(x)],[-.2,0,typeset(O)],[2,.3,typeset(A)],[5,.3,typeset(B)],[-.25,-7,typeset(C)],[.8,-5.3,typeset(theta)],[.3,-5,typeset(alpha)],[1,-.3,typeset(20)],[3.5,-.3,typeset(30)]},font=[DEFAULT,14]): plots:-display(p||(1..6),scaling=constrained,axes=none,view=[-1..5,-7..1]);
Figure 3.8.6(a) Representative sketch
Figure 3.8.6(b) implies that there is but a single absolute maximum, somewhere near x=30.
Numeric Solution
Control-drag the objective function θx.
Context Panel: Optimization≻Maximize (local)
tan−150/x−tan−120/x→maximize0.442911044073639015,x=31.6227766015368
The return consists of a list with two objects. The first object is the optimal value of the objective function; the second, a list of the parameter value giving this extreme value. At approximately 31.623 m from point O, angle θ ≐ 0.44291 radians (about 25.4°) is at a maximum.
Analytic Solution
Define the objective function θx
Control-drag θx=… Context Panel: Assign Function
θx=tan−150/x−tan−120/x→assign as functionθ
Obtain the critical number
Write the equation for the critical number; Press the Enter key.
Context Panel: Solve≻Solve
θ′x=0
−50x21+2500x2+20x21+400x2=0
→solve
x=1010,x=−1010
Apply the Second-Derivative test
Evaluate θ″ at the positive solution for x. Context Panel: Evaluate and Display Inline
θ″1010 = −32450010
Point C should be at a distance of x = 1010≐31.623 m from point O.
The maximum value of angle θ is θ1010 = arctan1210−arctan1510≐0.4429110437 radians, or about 25.4°.
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