Hexagonal prism $B$ is the image of hexagonal prism A after dilation by a scale factor of $\frac{1}{4}$. If the volume of hexagonal prism $B$ is $2 \mathrm{~m}^{3}$, find the volume of hexagonal prism A, the preimage.
Answer
\(\boxed{128 \mathrm{~m}^{3}}\) is the volume of hexagonal prism A.
Step 1 ：Given that hexagonal prism B is the image of hexagonal prism A after dilation by a scale factor of \(\frac{1}{4}\), and the volume of hexagonal prism B is \(2 \mathrm{~m}^{3}\).
Step 2 ：The ratio of the volumes of the two prisms is the cube of the scale factor, which is \(\left(\frac{1}{4}\right)^3 = \frac{1}{64}\).
Step 3 ：Let the volume of hexagonal prism A be \(V_A\). Then, \(V_A \times \frac{1}{64} = 2\).
Step 4 ：Solving for \(V_A\), we get \(V_A = 2 \times 64 = 128\).
Step 5 ：\(\boxed{128 \mathrm{~m}^{3}}\) is the volume of hexagonal prism A.
