Share Tool + Management Kdan Cloud Topic 4 Questions A= Tv Tv Ov Comment & Markup T Annotation FAX OCR Q Editor OCR Convert Fax Search Outlines Q Search Optimization Unconstrained optimization Constrained optimization (... (b) f(x,y)=(x-1)² + (y-2)²; constraint a² + y² = 9 82. Joan wants to fence a piece of land next to a river. The total surface of the fenced plot of land has to be 200 m². We will find the dimensions h and w (in m) so that the amount of fence needed is minimal, by three different methods. 195% h พ (a) Use the substitution method to find h and w (i.e., reduce the optimization problem of two variables with a constraint to an optimization problem of one variable by substitution). Does this method allow you to conclude that the dimensions you have obtained indeed correspond to a minimum? (b) Let S (h, w) denote the surface area of the fenced plot (in m²) and F (h, w) the amount of fence needed (in m). Graph the appropriate contour line of S (h, w) and a contour map of F (h, w) on the (h, w)-plane. Explain how you can estimate the answer to the question from the graph. Confirm your estimation by calculations. Does this method allow you to conclude that the dimensions you have obtained indeed correspond to a minimum? (c) Use the method of Lagrange to find h and w. Does this method allow you to conclude that the dimensions you have obtained indeed correspond to a minimum? What extra information does this method give you compared to the methods used in parts a and b? 4
Share Tool + Management Kdan Cloud Topic 4 Questions A= Tv Tv Ov Comment & Markup T Annotation FAX OCR Q Editor OCR Convert Fax Search Outlines Q Search Optimization Unconstrained optimization Constrained optimization (... (b) f(x,y)=(x-1)² + (y-2)²; constraint a² + y² = 9 82. Joan wants to fence a piece of land next to a river. The total surface of the fenced plot of land has to be 200 m². We will find the dimensions h and w (in m) so that the amount of fence needed is minimal, by three different methods. 195% h พ (a) Use the substitution method to find h and w (i.e., reduce the optimization problem of two variables with a constraint to an optimization problem of one variable by substitution). Does this method allow you to conclude that the dimensions you have obtained indeed correspond to a minimum? (b) Let S (h, w) denote the surface area of the fenced plot (in m²) and F (h, w) the amount of fence needed (in m). Graph the appropriate contour line of S (h, w) and a contour map of F (h, w) on the (h, w)-plane. Explain how you can estimate the answer to the question from the graph. Confirm your estimation by calculations. Does this method allow you to conclude that the dimensions you have obtained indeed correspond to a minimum? (c) Use the method of Lagrange to find h and w. Does this method allow you to conclude that the dimensions you have obtained indeed correspond to a minimum? What extra information does this method give you compared to the methods used in parts a and b? 4
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Share
Tool
+
Management Kdan Cloud
Topic 4 Questions
A=
Tv Tv Ov
Comment & Markup
T
Annotation
FAX
OCR
Q
Editor
OCR
Convert
Fax
Search
Outlines
Q Search
Optimization
Unconstrained optimization
Constrained optimization (...
(b) f(x,y)=(x-1)² + (y-2)²; constraint a² + y² = 9
82. Joan wants to fence a piece of land next to a river. The total surface of the fenced plot of land has
to be 200 m². We will find the dimensions h and w (in m) so that the amount of fence needed is
minimal, by three different methods.
195%
h
พ
(a) Use the substitution method to find h and w (i.e., reduce the optimization problem of two
variables with a constraint to an optimization problem of one variable by substitution). Does
this method allow you to conclude that the dimensions you have obtained indeed correspond to
a minimum?
(b) Let S (h, w) denote the surface area of the fenced plot (in m²) and F (h, w) the amount of fence
needed (in m). Graph the appropriate contour line of S (h, w) and a contour map of F (h, w)
on the (h, w)-plane. Explain how you can estimate the answer to the question from the graph.
Confirm your estimation by calculations. Does this method allow you to conclude that the
dimensions you have obtained indeed correspond to a minimum?
(c) Use the method of Lagrange to find h and w. Does this method allow you to conclude that the
dimensions you have obtained indeed correspond to a minimum? What extra information does
this method give you compared to the methods used in parts a and b?
4
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