Determine the volume of the solid in the 1 st octant bounded by $2 x+y+z=2$ Give your result as a reduced fraction. Submit Question

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 \(\boxed{\frac{6}{11}}\).