Find the volume of the tetrahedron bounded by the planes x + 2y + z = 2, x = 2y, x = 0, andz = 0

To find the volume of the tetrahedron, we first need to find the coordinates of the vertices where the planes intersect.

1. Intersection of planes x + 2y + z = 2, x = 2y, and x = 0: Substituting x = 0 in the first two equations, we get: 0 + 2y + z = 2 and 0 = 2y Solving these, we get y = 0 and z = 2. So, the coordinates of the first vertex are (0, 0, 2).

2. Intersection of planes x + 2y + z = 2, x = 2y, and z = 0: Substituting z = 0 in the first two equations, we get: x + 2y = 2 and x = 2y Solving these, we get x = 2 and y = 1. So, the coordinates of the second vertex are (2, 1, 0).

3. Intersection of planes x + 2y + z = 2, x = 0, and z = 0: Here, we directly get the coordinates of the third vertex as (0, 0, 0).

4. Intersection of planes x = 2y, x = 0, and z = 0: Here, we directly get the coordinates of the fourth vertex as (0, 0, 0).

Now, we have the coordinates of the four vertices of the tetrahedron: (0, 0, 2), (2, 1, 0), (0, 0, 0), and (0, 0, 0).

The volume V of a tetrahedron with vertices (x1, y1, z1), (x2, y2, z2), (x3, y3, z3), and (x4, y4, z4) is given by the formula:

V = 1/6 * |x1(y2z3 + y3z4 + y4z2 - y2z4 - y3z2 - y4z3) + x2(y3z4 + y4z1 + y1z3 - y3z1 - y4z3 - y1z4) + x3(y4z1 + y1z2 + y2z4 - y4z2 - y1z4 - y2z1) + x4(y1z2 + y2z3 + y3z1 - y1z3 - y2z1 - y3z2)|

Substituting the coordinates of the vertices in the formula, we get:

V = 1/6 * |0(00 + 02 + 20 - 02 - 00 - 20) + 2(02 + 10 + 00 - 00 - 12 - 00) + 0(00 + 00 + 02 - 00 - 02 - 00) + 0(00 + 00 + 00 - 00 - 00 - 00)|

Solving this, we get V = 1/6 * |0 + 2(0 - 2) + 0 + 0| = 1/6 * |-4| = 4/6 = 2/3 cubic units.

So, the volume of the tetrahedron is 2/3 cubic units.