Question

The volume of Solid A is 28 m and the volume of Solid B is 1,792 m. If the solids are similar, what is the ratio of the surface area of Solid A to thesurface area of Solid B?Surface Area Ratio:

Answer 1

For similar solids, the ratios of the two sides (R), two areas (A) and two volumes (V) are

`(R_1)/(R_2)\Rightarrow\text{Ratio of sides}`
`(A_1)/(A_2)=\frac{(R^{}_1)^2}{(R_2)^2}\Rightarrow\text{Ratio of surface areas}`
`\begin{gathered} (V_1)/(V_2)=((R_1)^3)/((R_2)^3) \\ 3\sqrt[]{(V_1)/(V_2)}=(R_1)/(R_2) \end{gathered}`

Therefore, we can relate the volumes as follows:

`\frac{V_1}{V_2_{}}=(28)/(1792)=(1)/(64)`

The ratio of the sides is given as

`(R_1)/(R_2)=3\sqrt[]{(1)/(64)}=(1)/(4)`

Therefore, the ratio of the surface areas is given as

`(A_1)/(A_2)=(1^2)/(4^2)=(1)/(16)`

Therefore, the ratio is 1:16.