Final answer:
To find the volume of the tetrahedron enclosed by the coordinate planes and the plane 2x+y+z=10, we need to find the height and the area of the base. The height is 10 units and the area of the base is 50 square units. Using the formula for the volume of a tetrahedron, the volume is 166.67 cubic units.
Step-by-step explanation:
The given solid is a tetrahedron enclosed by the coordinate planes and the plane 2x+y+z=10. To find its volume, we need to find the height and the area of the base. The height is the perpendicular distance between the base and the plane, which is given by the equation of the plane: 2x+y+z=10. When x=y=z=0, we have 0+0+0=10, so the height is 10. The base of the tetrahedron is the triangle formed by the intersection of the coordinate planes, which has sides of length 10.
To find the area of the base, we can use the equation of a triangle given its vertices. The vertices of the triangle are (0,0,0), (10,0,0), and (0,10,0). Using the formula for the area of a triangle, A=1/2 * base * height, we get A=1/2 * 10 * 10 = 50.
Finally, we can use the formula for the volume of a tetrahedron, V=1/3 * base area * height, to find the volume. Plugging in the values, we get V=1/3 * 50 * 10 = 166.67 cubic units.