The paraboloid z = x² + y² intersects the plane x+y+z=1 in the shape of an ellipse. The objective of this problem is to find the point on the ellipse closest and furtherest away from the origin. To this end, we establish the following constrained optimization problem: max./min. s.t. f(x, y, z)=√x² + y² + z² z = x² + y² x+y+z=1 To reduce the problem from three variables to two (i.e., removing z), substitute the first constraint into the objective function and square the result, then substitute the first constraint into the second constraint. • New objective: g(x, y) • New constraint: x² + y² +x+y=1 √x² + y² + x² + 2x²y² + y²4
The paraboloid z = x² + y² intersects the plane x+y+z=1 in the shape of an ellipse. The objective of this problem is to find the point on the ellipse closest and furtherest away from the origin. To this end, we establish the following constrained optimization problem: max./min. s.t. f(x, y, z)=√x² + y² + z² z = x² + y² x+y+z=1 To reduce the problem from three variables to two (i.e., removing z), substitute the first constraint into the objective function and square the result, then substitute the first constraint into the second constraint. • New objective: g(x, y) • New constraint: x² + y² +x+y=1 √x² + y² + x² + 2x²y² + y²4
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
The paraboloid z=x^2 + y^2 intersects the plane x+y+z=1x+y+z=1 in the shape of an ellipse.
The objective of this problem is to find the point on the ellipse closest and furtherest away from the origin. To this end, we establish the following constrained optimization problem:
max/min f(x,y,z)=√x^2+y^2+z^2
s.t. z=x^2+y^2
x+y+z=1
To reduce the problem from three variables to two (i.e., removing z), substitute the first constraint into the objective function and square the result, then substitute the first constraint into the second constraint.
Transcribed Image Text:The paraboloid z = x² + y² intersects the plane x+y+z= 1 in the shape of an ellipse.
The objective of this problem is to find the point on the ellipse closest and furtherest away from the
origin. To this end, we establish the following constrained optimization problem:
max./min.
s.t.
f(x, y, z) = √√√x² + y² + 2²
z = x² + y²
x+y+z=1
To reduce the problem from three variables to two (i.e., removing z), substitute the first constraint into
the objective function and square the result, then substitute the first constraint into the second
constraint.
• New objective: g(x, y)
• New constraint: x² + y² +x+y=1
We have now constructed an equivalent optimization problem.
+y² + x² + 2x²y²+y¹ x
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
