(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the equation of the plane through the point [1,2,2] that cuts off the smallest possible volume in the first octant.

2. Relevant equations

Volume of a pyramid = 1/3Ah

3. The attempt at a solution

The plane is going to cut out a pyramid with the x-, y-, and z-intercepts, so let x, y, and z be the intercepts. Then V = 1/6xyz. But since the plane must go through [1,2,2] and three points define a plane, we can write one of x, y, and z in terms of the other two. Any plane passing through intercepts x, y, and z has a general point [a,b,c] so that:

a/x + b/y + c/z = 1

Since [1,2,2] is on the plane, plug that in for [a,b,c]:

1/x + 2/y + 2/z = 1

Solve for x (just because I'm guessing that it would be the easiest):

x = -1/(2y + 2z - 1)

Now plug that into the volume formula:

V = -yz/(12y + 12z - 6)

Is this right so far? If not, what did I do wrong? If so, how can I continue?

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# Homework Help: Volume in first octant cut off by a plane

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