This problem involves two mathematically similar solid prisms, $A$ and $B$. Since the prisms are similar, their surface areas and volumes will follow specific proportional relationships.

Step 1: Finding the scale factor for surface area

Given that:

• The surface area of $A$ is $72.5 \, \text{cm}^2$
• The surface area of $B$ is $104.4 \, \text{cm}^2$

The surface areas of similar solids are proportional to the square of their scale factor. Let the scale factor be $k$. The ratio of the surface areas is:

$\frac{\text{Surface area of } B}{\text{Surface area of } A} = k^2$

Substituting the known values:

$\frac{104.4}{72.5} = k^2$

Solving for $k$:

$k^2 = \frac{104.4}{72.5} \approx 1.44$ $k = \sqrt{1.44} = 1.2$

Thus, the scale factor $k$ is $1.2$.

Step 2: Using the scale factor to find the volume of $B$

The volumes of similar solids are proportional to the cube of the scale factor. The volume of $A$ is given as $162.5 \, \text{cm}^3$. So, the volume of $B$, denoted as $V_B$, can be found using the formula:

$\frac{\text{Volume of } B}{\text{Volume of } A} = k^3$

Substituting the known values:

$\frac{V_B}{162.5} = (1.2)^3$ $V_B = 162.5 \times 1.728$ $V_B \approx 280.32 \, \text{cm}^3$

Final Answer:

The volume of $B$ is approximately $280.32 \, \text{cm}^3$.

Do you need more details or have any questions?

Related questions:

1. What is the relationship between the areas of two similar shapes?
2. How would the volume change if the scale factor were doubled?
3. Can the same principle be applied to non-prismatic shapes?
4. How can you calculate the height of the prisms given their volumes?
5. What are other geometric properties that change with the scale factor?

Tip: When dealing with similar solids, always remember that surface areas are proportional to the square of the scale factor, while volumes are proportional to the cube of the scale factor.