Chapter 3: Applications of Differentiation
Section 3.8: Optimization
A telecommunications company contemplates running a fiber-optic cable from point A to point C in the woods and along the road in Example 3.8.8. If the ratio of costs (woods/road) is λ>1, determine the characteristics of a minimal-cost solution to this problem of laying a communications cable.
Using Figure 3.8.8(a) in Example 3.8.8, the objective function (cost) can be written as
subject to the implied constraint x∈0,8.
Define the objective function cx
Context Panel: Assign Function
cx=λ x2+25+8−x→assign as functionc
Obtain the critical number
cx equal to zero.
Press the Enter key.
Context Panel: Solve≻Obtain Solutions for≻x
Context Panel: Select Element≻1
Context Panel: Assign to a Name≻Xmin
→solutions for x
→select entry 1
→assign to a name
Obtain the (relative) minimal cost
Evaluate cx at x=Xmin.
Context Panel: Simplify≻Symbolic
Context Panel: Assign to a Name≻Cmin
cXmin→simplify symbolic5⁢λ2+8⁢λ2−1−5λ2−1→assign to a nameCmin
Obtain the cost at the endpoints x=0 and x=8
Context Panel: Assign to a Name≻C0
c0→simplify symbolic5⁢λ+8→assign to a nameC0
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻C8
c8 = λ⁢89→assign to a nameC8
Comparison of Cmin, C0, and C8
Figure 3.8.9(a) compares Cmin with C0 and C8.
For all values of λ>1, Cmin gives the absolute minimum cost, but the graph of C8 is tangent to the graph of Cmin at λ=89/8≐1.18.
For λ<8/89−5≐1.8, C8<C0; otherwise, C8>C0.
The intersection of the graphs of C0 and C8 is the most surprising outcome of the investigation. For any relative costs, the combined path determined by Xmin gives the absolute minimal cost. But the relative standing of the endpoint costs depends on relationship of the costs in the woods and along the road.
Figure 3.8.9(a) Comparison of Cmin (black), C0 (red) and C8 (green)
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