Calculate Rectangular Prism Volume: 15cm Side Length with 3:1 Diagonal Ratio

To calculate the volume of the prism, let's use the given information and apply a series of logical steps:

Given data:
- $AB = 4$ cm
- $DB = 15$ cm
- $AD = 3 \times AC$

First, we note that $AC$ is a diagonal on the face $ABC$. Thus:
$AC = \sqrt{AB^2 + BC^2} = \sqrt{4^2 + BC^2}$

Now consider triangle $ACD$ where $A$ is one vertex of the rectangular prism and $D$ is opposite on the longest diagonal.
$AD = \sqrt{AB^2 + BD^2} = \sqrt{4^2 + 15^2} = \sqrt{16 + 225} = \sqrt{241}$

We substitute into the diagonal relation $AD = 3 \times AC$:
$\sqrt{241} = 3 \times \sqrt{16 + BC^2}$

Square both sides:
$241 = 9 \times (16 + BC^2)$

Solving for $BC^2$:
$241 = 144 + 9BC^2 \Rightarrow 97 = 9BC^2 \Rightarrow BC^2 = \frac{97}{9}$

Therefore, $BC = \sqrt{\frac{97}{9}} = \frac{\sqrt{97}}{3}$.

With the dimensions $AB = 4$, $BC = \frac{\sqrt{97}}{3}$, and $BD = 15$, we calculate the volume of the rectangular prism as:
$\text{Volume} = AB \times BC \times CD = 4 \times \frac{\sqrt{97}}{3} \times 15 \approx 300 \text{ cm}^3$

Thus, the volume of the rectangular prism is $\textbf{300 cm}^3$.