calculate the volume

Question

There is a triangular prism with infinite height. It has three edges parallel to z-axis, each passing through points $(0, 0, 0)$, $(3, 0, 0)$ and $(2, 1, 0)$ respectively. Calculate the volume within its side surfaces as well as the planes $z=2x+3y+6$ and $z=2x+7y+8$.

Answer 1

I don't necessarily see this as a calculus problem--more of a geometry problem (although I understand the calculus tag since vectors/planes tend to be taught in Cal-II or Cal-III). You need to find the area of the base (the triangle), this can be found from the three coordinates given using something like Heron's Formula (I don't know if that triangle permits an easier way or not--it would appear the triangle given is a right triangle). Then you need to find where the planes intersect the triangular prism. I suspect by the wording that the two planes do not intersect inside of the prism (otherwise the wording is poor and this is a more difficult problem).

Assuming the planes do not intersect inside of the prism, then you can break the region into three parts: 1) a right triangular prism, and 2) and 3) two pyramids on each "tip". Using the base area found from the three points and the orientation/intersection points of the two planes and the prism, you can find the height of the "middle" prism and the heights of the two pyramids and thus find the total volume.

Hint:

The lines of the three edges are given by:

1. $\vec{l}_1(t) = (x, y, z) = (0, 0, t)$
2. $\vec{l}_2(t) = (x,y,z) = (3,0,t)$
3. $\vec{l}_3(t) = (x, y, z) = (2,1,t)$

Find the intersections of each of the two planes with those three lines (so you should end up with two sets of three points). This will aid in identifying the three parts I alluded to above.

Answer 2

If you want to use a vector method, you could divide the prism into three tetrahedra and use the scalar triple product formula for the volume of a tetrahedron, which is $$V=\tfrac 16|\underline a.(\underline b \times \underline c)|$$

where the edge vectors $\underline a$, $\underline b$ and $\underline c$ share a common vertex