Question

Find the volume of the region between the planes x plus y plus z equals 1x+y+z=1 and 2 x plus 2 y plus z equals 22x+2y+z=2 in the first octant.

Answer 1

Answer:

  1/6 cubic unit

Step-by-step explanation:

Each plane describes the boundary of a triangular pyramid in the first octant. If we consider the x-y plane to be the base, that base will have an area that is 1/2 the product of the x- and y-intercepts. The height of the pyramid will be the z-intercept, so the volume is ...

  V = (1/3)Bh = (1/3)((1/2)(x-intercept)(y-intercept))(z-intercept))

  = (1/6)(x-intercept)(y-intercept)(z-intercept)

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We can write the equation of each plane in intercept form by dividing by the constant on the right:

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

  1st plane: x/1 + y/1 + z/1 = 1 . . . . . enclosed volume = (1/6)(1)(1)(1) = 1/6

  2nd plane: x/1 + y/1 + z/2 = 1 . . . enclosed volume = (1/6)(1)(1)(2) = 2/6

The difference between the volumes enclosed by the planes is the volume of the region between them, so is ...

  V = (2/6) -(1/6) = 1/6 . . . . cubic units

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We note that the x- and y-intercepts of both planes are the same, so the volume is that of a triangular wedge having an edge on the line x+y=1 in the plane z=0.

Answer 2

Answer:

1/6 cubic unit

Step-by-step explanation:

The volume of the region between the planes x plus y plus z equals 1x+y+z=1 and 2 x plus 2 y plus z equals 22x+2y+z=2 in the first octant is 1/6 cubic unit.