Find the closest point to the origin on the curve of intersection to a cone

In summary, the conversation discusses finding the point closest to the origin on the curve of intersection of a plane and a cone. The conversation also mentions using the equations gradF, gradH, and gradG, and setting up multiple constraint equations to solve for x, y, z, lambda, and mu. The conversation concludes by discussing the importance of having the constraints in the correct form for solving the problem.
  • #1
jimbo71
81
0

Homework Statement


find the point closest to the origin on the curve of intersection of the plane 2y+4z=5 and the cone z^2=4x^2+4y^2


Homework Equations





The Attempt at a Solution


see 40 attachement. I found the used f(x,y,z)=x^2+y^2+z^2 and found its gradient. found ggrad and hgrad and set fgrad=lambda*ggrad+mu*hgrad. using the two constraint equations i attempted to solve for x,y,z,lamdbda,mu. Either I messed up the setting up of this problem or my algebra is wrong some where becuase I keep getting x^2=-25. This is preventing me from solving x,y,z,lambda,mu. Please tell me which of the two possible mistakes I made and also what steps do I take after correctly solving for x,y,z,lambda,mu. Thank you!
 
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  • #2
without seeing your work, its hard to guess what you did wrong. there should be more than one case available. if you're getting nonsense in one case, you still have the other cases to check. once you get x,y,z, plug these into sqrt(f(x,y,z)) to get a distance. if they're ALL nonsense, then you did something wrong.

varify:
gradF=
gradH=
gradG=

gradF=L*gradH+M*gradH
constraint1=0
constraint2=0,
these give 5 equations in 5 unknowns.
 
  • #3
why is do you say constraint 1 and 2 are =0. i thought constraint 1 was =5 because it is 2y+5z and constraint 2=0 because you could subtract the z^2 over and get zero?
 
  • #4
well, we want the constraints in the form H(x,y,z)=0 G(x,y,z)=0 so that we can take the three dimensional gradient [d/dx,d/dy,d/dz] for the multiplier equation. you still haven't given anything for us to work with.
 
  • #5
In other words, "constraint 1" is H(x, y, z)= 2x+5z- 5= 0 and "constraint 2" is [itex]G(x,y,z)= z^2- 4x^2- 4y^2= 0[/itex].
 

1. What is the meaning of "closest point to the origin" in this context?

The "closest point to the origin" refers to the point on the curve of intersection between the cone and the plane that is closest to the origin (0,0,0) on the Cartesian coordinate system.

2. Why is finding the closest point to the origin on the curve of intersection important?

This calculation is important in various fields of science, such as engineering, physics, and mathematics. It can help determine the optimal location for a structure or object, as well as aid in solving optimization problems.

3. What is the mathematical formula for finding the closest point to the origin on the curve of intersection?

The mathematical formula for finding the closest point to the origin on the curve of intersection involves setting up and solving a system of equations. This can be done using methods such as Lagrange multipliers or substitution.

4. Are there any real-world applications for finding the closest point to the origin on the curve of intersection?

Yes, there are many real-world applications, including finding the optimal location for a satellite or antenna, determining the shortest path for a robot or vehicle, and solving optimization problems in economics or finance.

5. Are there any limitations or challenges in finding the closest point to the origin on the curve of intersection?

There can be limitations and challenges in certain scenarios, such as when the curve of intersection is complex or when the cone and plane intersect at multiple points. In these cases, alternative methods or approximations may be needed to find the closest point to the origin.

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