Chapter 3: Applications of Differentiation
Section 3.8: Optimization
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?
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
Figure 3.8.6(b) is a graph of θx.
f := arctan(50/x)-arctan(20/x);
Figure 3.8.6(b) Graph of θx
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.
Control-drag the objective function θx.
Context Panel: Optimization≻Maximize (local)
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.
Define the objective function 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
Apply the Second-Derivative test
Evaluate θ″ at the positive solution for x.
Context Panel: Evaluate and Display Inline
θ″1010 = −324500⁢10
Point C should be at a distance of x = 1010≐31.623 m from point O.
The maximum value of angle θ is θ1010 = arctan⁡12⁢10−arctan⁡15⁢10≐0.4429110437 radians, or about 25.4°.
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