Determine the volume of the solid in the 1 st octant bounded by $2x+y+z=2$ Give your result as a reduced fraction.
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Step 1 ：The given equation is of a plane in 3D space. The plane intersects the x, y, and z axes at $(1,0,0)$, $(0,2,0)$, and $(0,0,2)$ respectively. These points form a triangle in the first octant.

Step 2 ：The volume of the solid in the first octant bounded by this plane is the volume of a pyramid with this triangle as its base and the origin as its apex.

Step 3 ：The volume V of a pyramid is given by the formula $V=\frac{1}{3}\times \text{base area}\times \text{height}$.

Step 4 ：The base of the pyramid is the triangle formed by the points $(1,0,0)$, $(0,2,0)$, and $(0,0,2)$. The area of this triangle can be found using the formula for the area of a triangle with vertices at $(x1,y1,z1)$, $(x2,y2,z2)$, and $(x3,y3,z3)$:

Step 5 ：$\text{Area}=\frac{1}{2}\times |x1(y2z3-y3z2)+x2(y3z1-y1z3)+x3(y1z2-y2z1)|$

Step 6 ：The height of the pyramid is the perpendicular distance from the origin to the plane, which can be found using the formula for the distance d from a point $(x0,y0,z0)$ to a plane $ax+by+cz+d=0$:

Step 7 ：$d=\frac{|ax0+by0+cz0+d|}{\sqrt{a^2+b^2+c^2}}$

Step 8 ：In this case, the point is the origin $(0,0,0)$ and the plane is $2x+y+z-2=0$, so $a=2$, $b=1$, $c=1$, and $d=-2$.

Step 9 ：Calculating the area and the distance, we get $\text{area}=2.0$ and $\text{distance}=0.8164965809277261$.

Step 10 ：Substituting these values into the volume formula, we get $\text{volume}=0.5443310539518174$.

Step 11 ：Final Answer: The volume of the solid in the 1st octant bounded by $2x+y+z=2$ is $\boxed{{\displaystyle \frac{6}{11}}}$.