Question

Find the scale factor of the two prisms, the ratio of their surface areas, and the ratio of theirvolumes. List the larger values first.6 cm8 cm9 cm10 cm12 cm15 cmScale FactorSurface AreasVolumesBlank 1:Blank 2:Blank 3:Blank 4:Blank 5:Blank 6:

Answer 1

We will operate as follows:

*Scale factor:

We determine the scale factor using two respective sides, that is:

`9x=6\Rightarrow x=(6)/(9)\Rightarrow x=(2)/(3)`

So, the scale factor is 2 : 3.

*Surface area:

We determine the surface are of each prism:

`S_(A1)=(9\cdot12)+2((15\cdot9)/(2))+(12\cdot\sqrt[]{15^2+9^2})+(15\cdot12)\Rightarrow S_(A1)=(108)+(135)+(36\sqrt[]{34})+(180)`
`\Rightarrow S_(A1)=423+36\sqrt[]{34}\Rightarrow S_(A1)=632.9142682\ldots`
`S_(A2)=(6\cdot8)+2((6\cdot10)/(2))+(10\cdot8)+(8\cdot\sqrt[]{6^2+10^2})\Rightarrow S_(A2)=(48)+(60)+(80)+(16\sqrt[]{34})`
`\Rightarrow S_(A2)=188+16\sqrt[]{34}\Rightarrow S_(A2)=281.2952303\ldots`

Now:

`(423+36\sqrt[]{34})x=188+16\sqrt[]{34}\Rightarrow x=\frac{188+16\sqrt[]{34}}{423+36\sqrt[]{34}}\Rightarrow x=(4)/(9)`

So, the ratio of the surface areas is 4 : 9.

*Volume:

We determine the volume of each prism and proceed as before:

`V_1=(9\cdot15\cdot12)/(2)\Rightarrow V_1=810`
`V_2=(6\cdot10\cdot8)/(2)\Rightarrow V_2=240`

Now:

`810x=240\Rightarrow x=(240)/(810)\Rightarrow x=(8)/(27)`

So, the ratio for the volumes is 8 : 27.