Find eqn of plane thru pt (1,4,3) cutting off smallest vol.

[MarkSA]

Hi,

Q: Find an equation of the plane that passes through the point (1,4,3) and cuts off the smallest volume in the first octant.

Is there an easy way to do this problem that does not involve 10 billion miles of algebra, derivatives, and lots of dead trees? It's on my homework and I decided i'd look at the solution manual first since i've chewed up 3 hours already on it. The problem is so long and tedious it brings tears to my eyes. I'm wondering if there is some easy trick to this one that does not use the volume of a tetrahedron to solve it...

 

[Deleted member 4993]

Re: Is there an easy way to do this...

Hi,

Q: Find an equation of the plane that passes through the point (1,4,3) and cuts off the smallest volume in the first octant.

Is there an easy way to do this problem that does not involve 10 billion miles of algebra, derivatives, and lots of dead trees? It's on my homework and I decided i'd look at the solution manual first since i've chewed up 3 hours already on it. The problem is so long and tedious it brings tears to my eyes. I'm wondering if there is some easy trick to this one that does not use the volume of a tetrahedron to solve it...

The problem is I have no idea what method will take 10 billion miles of algebra. I am sure you are exaggerating - but I don't know what to expect.

If I were to do this problem,

I would assume that the plane passes through (x[sub:2wmr19d6]1[/sub:2wmr19d6], 0, 0), (0, y[sub:2wmr19d6]1[/sub:2wmr19d6],0) & (1,4,3)

Then I'll find the point (0,0,z[sub:2wmr19d6]1[/sub:2wmr19d6]) and express z[sub:2wmr19d6]1[/sub:2wmr19d6] in terms of x[sub:2wmr19d6]1[/sub:2wmr19d6] & y[sub:2wmr19d6]1[/sub:2wmr19d6]

Now I'll find the volume in terms of x[sub:2wmr19d6]1[/sub:2wmr19d6] & y[sub:2wmr19d6]1[/sub:2wmr19d6]

Now I'll minimize volume interms of x[sub:2wmr19d6]1[/sub:2wmr19d6] & y[sub:2wmr19d6]1[/sub:2wmr19d6]

Judicious use of text book and google will reduce your work. For example for volume of a tetrahedra:

For a tetrahedron with vertices a = (a1, a2, a3), b = (b1, b2, b3), c = (c1, c2, c3), and d = (d1, d2, d3), the volume is (1/6)·|det(a?b, b?c, c?d)|,

It becomes lot simpler if the origin is one of the vertices and the edges are the axes of the co-ordinate system.
[ref: http://en.wikipedia.org/wiki/Tetrahedron]

Instead of trees, you need to spend some electrons in the virtual space.....