# Homework Help: Maximize a sphere

1. Jun 21, 2009

### davedave

1. The problem statement, all variables and given/known data

Consider the tetrahedron in the FIRST octant defined by x+y+z/2=1.
Find the maximum sphere inside the tetrahedron.

2. Relevant equations

I use Lagrange Multipliers. let L be lamba.

(del)f(x,y,z)=L*(del)g(x,y,z)

3. The attempt at a solution

I don't know if I can assume that the center of the sphere is (a,a,2a) where 0<a<1

reason: Since the x, y, z intercepts of the tetrahedron are 1, 1, 2 respectively, I let the z
coordinate of the sphere be twice the x and y coordinates.

f(x,y,z)=(x-a)^2+(y-a)^2+(z-2a)^2

g(x,y,z)=x+y+z/2-1=0

next, take the gradient of f and g in the equation

2(x-a)i+2(y-a)j+2(z-2a)k=L*(i+j+k/2)

solving for x, y, z gives x=L/2+a y=L/2+a z=L/4+2a

put them into the tetrahedron equation and solve for lamba L

L=8/9 * (1-3a)

now put the value of L into the x, y, z equations which gives

x=4/9*(1-3a)+a y=4/9*(1-3a)+a z=2/9*(1-3a)+2a

thus, put those equations above into f(x,y,z)= 4/9*(1-3a)^2