We will operate as follows:
*Scale factor:
We determine the scale factor using two respective sides, that is:
[tex]9x=6\Rightarrow x=\frac{6}{9}\Rightarrow x=\frac{2}{3}[/tex]So, the scale factor is 2 : 3.
*Surface area:
We determine the surface are of each prism:
[tex]S_{A1}=(9\cdot12)+2(\frac{15\cdot9}{2})+(12\cdot\sqrt[]{15^2+9^2})+(15\cdot12)\Rightarrow S_{A1}=(108)+(135)+(36\sqrt[]{34})+(180)[/tex][tex]\Rightarrow S_{A1}=423+36\sqrt[]{34}\Rightarrow S_{A1}=632.9142682\ldots[/tex][tex]S_{A2}=(6\cdot8)+2(\frac{6\cdot10}{2})+(10\cdot8)+(8\cdot\sqrt[]{6^2+10^2})\Rightarrow S_{A2}=(48)+(60)+(80)+(16\sqrt[]{34})[/tex][tex]\Rightarrow S_{A2}=188+16\sqrt[]{34}\Rightarrow S_{A2}=281.2952303\ldots[/tex]Now:
[tex](423+36\sqrt[]{34})x=188+16\sqrt[]{34}\Rightarrow x=\frac{188+16\sqrt[]{34}}{423+36\sqrt[]{34}}\Rightarrow x=\frac{4}{9}[/tex]So, the ratio of the surface areas is 4 : 9.
*Volume:
We determine the volume of each prism and proceed as before:
[tex]V_1=\frac{9\cdot15\cdot12}{2}\Rightarrow V_1=810[/tex][tex]V_2=\frac{6\cdot10\cdot8}{2}\Rightarrow V_2=240[/tex]Now:
[tex]810x=240\Rightarrow x=\frac{240}{810}\Rightarrow x=\frac{8}{27}[/tex]So, the ratio for the volumes is 8 : 27.