Answer :
The problem involves finding the volume of the region bounded by the coordinate planes and the planes z = x + y and z = 10. The options provided are 250, 500, 1000/3, and 500/3.
To find the volume of the region, we need to determine the boundaries in the x, y, and z directions. The region is bounded by the coordinate planes (x = 0, y = 0, and z = 0) and the two given planes (z = x + y and z = 10).
To determine the boundaries in the x and y directions, we set the equations z = x + y and z = 10 equal to zero and solve for x and y. Setting z = x + y equal to zero, we have x + y = 0, which represents the line where the planes intersect the xy-plane. Thus, the boundary in the x and y directions is the line x + y = 0.To determine the boundaries in the z direction, we set the equation z = x + y equal to 10, resulting in x + y = 10. This represents the plane z = 10, which is parallel to the xy-plane. Thus, the boundary in the z direction is the plane z = 10.
The region bounded by these planes and coordinate planes forms a triangular pyramid. The height of the pyramid is the distance between the plane z = x + y and z = 10, which is 10 units. The base of the pyramid is the triangle formed by the line x + y = 0 in the xy-plane.The formula to calculate the volume of a pyramid is V = (1/3) * base area * height. In this case, the base area is given by the formula for the area of a triangle: (1/2) * base * height.
Since the base of the pyramid is a triangle with a base length of 10 units and a height of 10 units, the base area is (1/2) * 10 * 10 = 50 square units.
Substituting the values into the volume formula, we have:
V = (1/3) * 50 * 10 = 500/3 cubic units.
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