Question

Let R= [0, 1] x [0, 1]. Find the volume of the region above R and below the plane which passes through the three points (0,0,1), (1, 0,5) and (0, 1,9).

Answer 1

To find the volume of the region above R and below the plane passing through the three points (0,0,1), (1,0,5), and (0,1,9), we can use the formula for the volume of a parallelepiped.

Step-by-step explanation:

To find the volume of the region above R and below the plane passing through the three points (0,0,1), (1,0,5), and (0,1,9), we can use the formula for the volume of a parallelepiped. The three points form the vertices of the parallelepiped, and the volume of the parallelepiped is the absolute value of the triple product of the three vectors formed by the points.

The three vectors are (1,0,4), (0,1,8), and (1,0,4) - (0,1,8) = (1,-1,-4). The volume can be calculated as |(1,0,4) * (0,1,8) * (1,-1,-4)|, which is equal to 32.

Since the region above R and below the plane is half of the parallelepiped, the volume is 16.