Find the volume bounded by the paraboloid z 2x2 y2 and the

Solution

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)