Chapter 3: Applications of Differentiation
Section 3.8: Optimization
Example 3.8.10
The strength of fields such as gravity or electromagnetism is inversely proportional to distance from the source. If two such sources are separated by a distance of L units, determine where on the line connecting the sources will the minimum strength be detected? Assume that the strength of one source is k times that of the other.
Solution
Analysis
Let one source at x=0 on a number line have strength 1; and the other at x=L, have strength k.
Except for a proportionality constant, the combined strength at x is given by
Sx=1x2+kL−x2
Figure 3.8.10(a) is a representative graph of Sx with L=10 and k=5.
Figure 3.8.10(a) Field strength, L=10,k=5
Without loss of generality, L can be set to 1 so that Sx=1x2+k1−x2.
Analytic Solution
Define the objective function Sx
Control-drag Sx=… Context Panel: Assign Function
Sx=1x2+k1−x2→assign as functionS
Obtain the critical number
Write the equation S′x=0 and press the Enter key.
Context Panel: Solve≻Obtain Solutions for≻x
S′x=0
−2x3+2⁢k1−x3=0
→solutions for x
k2/3k+1−k1/3k+1+1k+1,−12⁢k2/3k+1+12⁢k1/3k+1+1k+1+12⁢I⁢3⁢k2/3k+1+k1/3k+1,−12⁢k2/3k+1+12⁢k1/3k+1+1k+1−12⁢I⁢3⁢k2/3k+1+k1/3k+1
Control-drag the first (and only real) solution. Context Panel: Assign to a Name≻Xmin
k2/3k+1−k1/3k+1+1k+1→assign to a nameXmin
Apply the Second-Derivative test
S″Xmin = 6k2/3k+1−k1/3k+1+1k+14+6⁢k1−k2/3k+1+k1/3k+1−1k+14
The even powers in the denominators of S″Xmin imply that the expression is positive, and hence that Xmin is indeed a relative minimum. Figure 3.8.10(a) is invoked as proof that Xmin is a global minimum.
The graph of Xmin as a function of the relative strength k in Figure 3.8.10(b) suggests computing
limk→0Xmin = 1 and limk→∞Xmin = 0
As the strength of the source on the right decreases, the minimum of S moves toward x=1.
As the strength of the source on the right increases, the minimum of S moves toward x=0.
Figure 3.8.10(b) Graph of Xmin as a function of k
<< Previous Example Section 3.8 Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document