Question

Find the volume of the pyramid with base in the plane z=−8 and sides formed by the three planes y=0 and y−x=3 and x+2y+z=3

Answer 1

To calculate the pyramid's volume, locate the base within the plane z = -8, determine its area by solving the intersections of the given planes, then find the height and apply the volume formula for pyramids: V = (1/3) * base area * height.

Step-by-step explanation:

To find the volume of the pyramid with a base in the plane z = -8 and sides formed by the planes y = 0, y - x = 3, and x + 2y + z = 3, we first need to determine the area of the base and the height of the pyramid.

The base of the pyramid lies in the plane z = -8, so we must find the intersection points of this plane with the other two planes within the z = -8 plane to determine the shape and area of the base.

After solving these intersections and finding the area of the base, we can find the height of the pyramid. The height is the perpendicular distance from the base plane z = -8 to the apex formed by the intersection of the three planes. Once we have both the area of the base and the height, we can use the formula for the volume of a pyramid: V = (1/3) * base area * height.

The calculations require solving a set of linear equations to find the vertices of the base and then using geometry or linear algebra to find the area and height.

Answer 2

To find the volume of the pyramid, we need to determine the area of the base and the height. The base is a triangle formed by three intersecting planes. The height is the perpendicular distance between the base and another plane. The volume is 30 unit³

Step-by-step explanation:

To find the volume of the pyramid, we need to determine the area of the base and the height. The base is formed by the intersection of the planes y=0, y-x=3, and x+2y+z=3. Solving these equations, we find that the base is a triangle with vertices (0,0,-3), (0,3,-6), and (3,3,-12).

The height of the pyramid can be found as the perpendicular distance from the plane z=-8 to the base. The distance between these two planes is 8-(-12) = 20.

Now we can use the formula for the volume of a pyramid: V = (1/3) * base area * height. The base area of the triangle can be found using the formula for the area of a triangle: A = (1/2) * base * height.

Calculating the values, we have base area = (1/2) * 3 * 3 = 4.5 and height = 20. Plugging these values into the volume formula, we get V = (1/3) * 4.5 * 20 = 30.