Solving Multivariable Calculus Final Problems: F. and G.

[CookieSalesman]

Homework Statement

I had a rather tough multivariable calculus final, and the class was creamed by it. Apparently the professor thinks a 40% average is okay.

But here is a portion of the problems. Please help me solve 'em. I'm not exactly sure if I'm taking the right path in these, but I did work on them to try and solve it.

F.

show that

G. xyz=a3 encloses a tetrahedron with any of its tangent planes. Show that the area of the tetrahedron is constant.
In my work, I sometimes just considered a3 to be any constant "c" for simplicity's sake.

Homework Equations

(Look... there's a ton of relevant equations..)

The Attempt at a Solution

Here is my attempt at a solution.

F.
http://postimg.org/image/fif8ixl53/
I'm actually pretty happy with this solution, seeing as I may have gotten it somewhere. But as for really figuring it out entirely I couldn't. I was just stuck at y/x and I can't really see how the equation works out.
http://postimg.org/image/xudxf024b/
There's my attempt at the end. I'm really not sure if this was mathematically allowed.

G.
http://postimg.org/image/o2myzhgw9/
Here's my attempt at this...
I really don't know what to do. It feels really simple, but at the time I tried I just couldn't do it.
I know you can represent the area of a tet by multiplying xy(.5) and then z(.5) which gives the area but I ended up not being able to do that. I think. I couldn't figure out how to write the equations. You can see those equations in the bottom left- they're boxed. I don't know if those are right. I'm not sure if the denominator is interchangable with the other remaining variable. I tried to draw level curves with one variable fixed at 1 in those 3 graphs, but I'm not sure if those were "right". Etc etc...
Thanks.

 

[Ray Vickson]

Homework Statement

I had a rather tough multivariable calculus final, and the class was creamed by it. Apparently the professor thinks a 40% average is okay.

But here is a portion of the problems. Please help me solve 'em. I'm not exactly sure if I'm taking the right path in these, but I did work on them to try and solve it.

F.

show that

G. xyz=a3 encloses a tetrahedron with any of its tangent planes. Show that the area of the tetrahedron is constant.
In my work, I sometimes just considered a3 to be any constant "c" for simplicity's sake.

Homework Equations

(Look... there's a ton of relevant equations..)

The Attempt at a Solution

Here is my attempt at a solution.

F.
http://postimg.org/image/fif8ixl53/
I'm actually pretty happy with this solution, seeing as I may have gotten it somewhere. But as for really figuring it out entirely I couldn't. I was just stuck at y/x and I can't really see how the equation works out.
http://postimg.org/image/xudxf024b/
There's my attempt at the end. I'm really not sure if this was mathematically allowed.

G.
http://postimg.org/image/o2myzhgw9/
Here's my attempt at this...
I really don't know what to do. It feels really simple, but at the time I tried I just couldn't do it.
I know you can represent the area of a tet by multiplying xy(.5) and then z(.5) which gives the area but I ended up not being able to do that. I think. I couldn't figure out how to write the equations. You can see those equations in the bottom left- they're boxed. I don't know if those are right. I'm not sure if the denominator is interchangable with the other remaining variable. I tried to draw level curves with one variable fixed at 1 in those 3 graphs, but I'm not sure if those were "right". Etc etc...
Thanks.

I would be willing to look at the solutions if you typed them out.