The problem involves a geometric figure resembling a trapezoidal prism and asks to compare the volumes of two specific shapes: the prism $P.Q.R.S.A.B.E.D$ and the triangle-based prism $A.B.C.D.E.F$.

Key Details:

1. Height AB is given as $2 \, \text{m}$.
2. $\angle ACB = \theta$, with $\tan \theta = 1$, which implies $\theta = 45^\circ$.
3. The ratio of the lengths $|PA| : |AC| = 5:2$.

The task is to find the ratio of the volumes of these two prisms, focusing on the geometric relationships between the shapes.

Step-by-Step Approach:

1. Prism ABC.DEF: The prism seems to have a triangular base $\triangle ABC$.

   • The height AB is $2 \, \text{m}$.
   • Since $\tan \theta = 1$, the base $\triangle ABC$ forms a right triangle with equal sides $AC$ and $BC$.
   • The volume of this triangular prism is the area of the base $\triangle ABC \times$ the height $DE$.

2. Prism PQRS.ABED: This figure seems to be a larger trapezoidal prism. Given that the ratio of $PA$ to $AC$ is $5:2$, this indicates that the dimensions of this prism are scaled up compared to $ABC.DEF$.

3. Volume Comparison: The volume ratio will depend on the ratio of the areas of their respective bases and the corresponding heights.

Let me compute the volumes and their ratio.

The ratio of the volumes of the two prisms, $P.Q.R.S.A.B.E.D$ and $A.B.C.D.E.F$, is 6.25. This means the volume of the larger prism $P.Q.R.S.A.B.E.D$ is 6.25 times the volume of the smaller prism $A.B.C.D.E.F$.

Would you like further details or have any questions?

Here are 5 related questions:

1. How is the volume of a triangular prism calculated?
2. What is the significance of the scaling factor in volume comparisons?
3. Why does the height cancel out when comparing the volumes?
4. How do geometric ratios affect area and volume?
5. What are other examples of shapes with similar volume relationships?

Tip: When scaling geometric shapes, remember that areas scale by the square of the linear dimension, and volumes scale by the cube of the linear dimension.