A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing


  • 5 years ago

Im not sure how to do this one: The temperature on the surface given by x^2 + y^2 + 4z^2 = 12 is given by T(x, y, z) = y^2 + 4z^2. What is the maximum temperature on this surface?

  • This Question is Closed
  1. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    You should use Lagrange multipliers to find the maximum temperature on the surface (which is an ellipsoid). The function is \[T(x,y,z)=y^2+4z^2.\] The constraint is \[g(x,y,z)=x^2+y^2+4z^2=12.\] The auxiliary function is \[\Lambda(x,y,z,\lambda)=T(x,y,z)+\lambda [g(x,y,z)-12]=\] \[=y^2+4z^2+\lambda[x^2+y^2+4z^2-12].\] Then you have to solve the following system of linear equations: \[\begin{array}{rcl} \frac{\partial\Lambda}{\partial x}=2x&=&0\\ \frac{\partial\Lambda}{\partial y}=2y+2\lambda y&=&0\\ \frac{\partial\Lambda}{\partial z}=8z+8\lambda z&=&0\\ \frac{\partial\Lambda}{\partial\lambda}=x^2+y^2+4z^2-12&=&0\\ \end{array}\] The solution is \[x=0,\quad \lambda=-1,\quad\textrm{ and you'll get: }y^2+4z^2=12.\] Since \[T(x,y,z)=y^2+4z^2\] therefore \[T(0,y,z)=12,\quad \textrm{ if } y=\pm\sqrt{12-4z^2},\quad -\sqrt{3}\leq z\leq \sqrt{3}.\] So the maximum temperature on the ellipsoid is 12 and the critical point are on the ellipse(!) \[y^2+4z^2=12.\]

  2. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...


  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.