
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.
$\displaystyle f(x,y,z)=(x10)^{2}+(y+4)^{2}+(z9)^{2}$
$\displaystyle {\nabla}f(x,y,z)=2(x10)i+2(y+4)j+2(z9)k$
Subject to the constraint $\displaystyle 7x+8y10z4=0$
$\displaystyle {\lambda}{\nabla}g(x,y,z)={\lambda}(7i+8j10k)$
So, $\displaystyle 2(x10)i+2(y+4)j+2(z9)k={\lambda}(7i+9j10k)$
Which leads to the equation:
$\displaystyle 2(x10)=7{\lambda}, \;\ 2(y+4)=8{\lambda}, \;\ 2(z9) = 10{\lambda}$
So, we have:
$\displaystyle {\lambda}=\frac{2(x10)}{7}, \;\ {\lambda}=\frac{y+4}{4}, \;\ {\lambda}=\frac{(z9)}{5}$
From the first two equations we get:
$\displaystyle \frac{2(x10)}{7}=\frac{y+4}{4}$
where $\displaystyle y=\frac{8x}{7}\frac{108}{7}$
From the first and third we get:
$\displaystyle \frac{2(x10)}{7}=\frac{(z9)}{5}$
where $\displaystyle z=\frac{10x}{7}+\frac{163}{7}$
Sub these into the constraint and get:
$\displaystyle \frac{213x}{7}\frac{2522}{7}=0$
Solving for x we get $\displaystyle x=\frac{2522}{213}$
Which leads to $\displaystyle y=\frac{404}{213}, \;\ z=\frac{1357}{213}$
Sub these into g(x,y,z):
$\displaystyle \sqrt{(x10)^{2}+(y+4)^{2}+(z9)^{2}}$
We get $\displaystyle \boxed{\frac{56}{\sqrt{213}}}$
Check it with the distance formula:
$\displaystyle \frac{7(10)+8(4)10(9)4}{\sqrt{7^{2}+8^{2}+(10)^{2}}}=\boxed{\frac{56}{\sqrt{213}}}$
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.