Minimizing Volume in 3-Space: Plane Equation?

[cragar]

I once had a home work question that asked us to find the equation of a plane that went through the point (1,1,1) an enclosed the least amount of volume in the first octant . I know how to do it with derivatives and all that but what if the plane was on edge going from (1,1,1) to the origin . It would be like having a sheet of glass bisect the corner of the room but my teacher said that it had no top on it and it enclosed no volume , so what do you guys think ?
I think the equation of my plane would be y=x in 3-space .

 

[HallsofIvy]

I think your teacher is completely correct! There are, in fact, an infinite number of planes passing through both (0, 0, 0) and (1, 1, 1) (y= x is one of them but rotating that around the line x= t+ 1, y= t+ 1, z= t+ 1 gives another for every angle of rotation between 0 and $2\pi$) but none of then "cut off" a bounded region of the first octant.

 

[cragar]

why can't we say it cuts off zero volume .

 

[torquil]

Depending on the detailed wording of the problem (and the chosen definition of the word 'enclose'), you probably could.

But the best would be to solve the problem both ways, i.e. also assume that you're supposed to find the plane that cuts off the minimal volume in that octant, while intersecting somewhere on all three axes.

Then you'd have your "clever solution" as well as what is probably the "intended solution".

 

[cragar]

ya i was thinking it would work