
Lagrange multiplier
1)Use Lagrange multipliers to find shortest distance from (10,4,9) to the plane 7x+8y10z=4
h=lambda(backwards h)
 I tried to use d=sqrt((x10)^2+(y+4)^2+(z9)^2) then split each one up to 2(x10)=7h , 2(y+4)=8h , 2(z9)=10h
 I then solved for x,y,z and plugged into the original equation of 7x+8y10z=4 got I got h=112/213 which is not correct and I'm not sure where I messed up.
2)Use lagrange multipliers to find volume of largest rectangular box in first octant with 3 faces in coordinate planes and one vertex in plane x+5y+2z=30.
I use V=xyz and got yz=h , xz=5h , xy=2h , x+5y+2z=30
 Not sure where to go after this.

For the first one, you can easily check your answer by using the distance from a point to a plane formula. But first....the long way.
Subject to the constraint
So,
Which leads to the equation:
So, we have:
From the first two equations we get:
where
From the first and third we get:
where
Sub these into the constraint and get:
Solving for x we get
Which leads to
Sub these into g(x,y,z):
We get
Check it with the distance formula:
Check....

worked out perfectly. When I plugged x,y,z into distance I got (56*sqrt(213)) / 213. Thanks for the steps that helped me a lot.