Pyramid intersections

Question

Let P be a square pyramid whose base consists of the four vertices (0,0,0),(3,0,0),(3,3,0), and (0,3,0), and whose apex is the point (1,1,3). Let Q be a square pyramid whose base is the same as the base of P, and whose apex is the point (2,2,3). Find the volume of the intersection of the interiors of P and Q.

Answer 1

Step 1: Define the Base and Apex of the Pyramids

1. The base of both pyramids is the square with vertices at (0,0,0)(0, 0, 0)(0,0,0), (3,0,0)(3, 0, 0)(3,0,0), (3,3,0)(3, 3, 0)(3,3,0), and (0,3,0)(0, 3, 0)(0,3,0).
2. The apex of pyramid PPP is at (1,1,3)(1, 1, 3)(1,1,3), and the apex of pyramid QQQ is at (2,2,3)(2, 2, 3)(2,2,3).

Step 2: Visualize the Pyramids

The pyramids PPP and QQQ share the same square base, which lies on the xyxyxy-plane. The difference between the pyramids lies in the location of their apexes:

1. Apex of PPP: (1,1,3)(1, 1, 3)(1,1,3)
2. Apex of QQQ: (2,2,3)(2, 2, 3)(2,2,3)

Since the apexes are at different positions, the pyramids intersect in a region above the shared base.

Step 3: Symmetry and Properties of Pyramids

The pyramids are symmetric with respect to the center of their shared base. Pyramid PPP has its apex shifted towards (1,1)(1, 1)(1,1), while pyramid QQQ's apex is shifted towards (2,2)(2, 2)(2,2).

Step 4: Geometry of the Intersection

The apexes of pyramids PPP and QQQ are relatively close to each other, and their heights are the same (both are at z=3z = 3z=3). Therefore, the intersection of the pyramids will form another pyramid whose base is a subset of the original square base and whose apex is somewhere between the apexes of PPP and QQQ.

Step 5: Volume of Pyramids and Intersection

The formula for the volume of a pyramid is:

V=13×Base Area×HeightV = \frac{1}{3} \times \text{Base Area} \times \text{Height}V=31​×Base Area×Height

For both pyramids PPP and QQQ, the base area is 999 (since the side length of the square base is 3), and the height is 3. Therefore, the volume of each pyramid is:

VP=VQ=13×9×3=9V_P = V_Q = \frac{1}{3} \times 9 \times 3 = 9VP​=VQ​=31​×9×3=9

Step 6: Estimate the Volume of Intersection

The volume of the intersection of two pyramids is typically a fraction of the volume of either pyramid. Due to the symmetry and closeness of the apexes, the intersection forms a pyramid-like region with a smaller base and the same height.

By geometric reasoning, the volume of the intersection can be estimated as half of the volume of one of the pyramids because the apexes are symmetrically positioned and the height is the same.

Thus, the volume of the intersection is:

Vintersection=12×9=4.5V_{\text{intersection}} = \frac{1}{2} \times 9 = 4.5Vintersection​=21​×9=4.5

Final Answer:

The volume of the intersection of the interiors of pyramids PPP and QQQ is 4.5\boxed{4.5}4.5​.