A solid sphere and a solid hemisphere have the same total surface area. Find the ratio of their volumes.

Hint: To find the ratio of volumes, we equate the surface areas of both the sphere and the hemisphere and then we find the relation between their radii. Then we calculate their volumes and find the ratio.


Complete step-by-step answer:
Let R be the radius of the solid sphere
Let r be the radius of the solid hemisphere.

We know the surface area of the solid sphere is $ {\text{4}}\pi {{\text{R}}^2} $ .
And we know, the surface area of the solid hemisphere is $ {\text{3}}\pi {{\text{r}}^2} $ .

Given Data, The surface areas of the sphere and hemisphere are equal.
 $
   \Rightarrow {\text{4}}\pi {{\text{R}}^2} = 3\pi {{\text{r}}^2} \\
   \Rightarrow {\text{4}}{{\text{R}}^2} = 3{{\text{r}}^2} \\
   \Rightarrow {\text{R = }}\dfrac{{\sqrt 3 {\text{r}}}}{2}{\text{ - - - - }}\left( 1 \right) \\
 $

We know, Volume of the solid sphere = $ \dfrac{4}{3}\pi {{\text{R}}^3} $
We know, Volume of the solid hemisphere = $ \dfrac{2}{3}\pi {{\text{r}}^3} $

The ratio of volumes = $ \dfrac{{\dfrac{4}{3}\pi {{\text{R}}^3}}}{{\dfrac{2}{3}\pi {{\text{r}}^3}}} $
                                      = $ \dfrac{{2{{\text{R}}^3}}}{{{{\text{r}}^3}}} $
Now we substitute the value of R from equation 1, we get
Ratio = $ \dfrac{{2{{\left( {\dfrac{{\sqrt 3 {\text{r}}}}{2}} \right)}^3}}}{{{{\text{r}}^3}}} $
          = $ \dfrac{{2\left( {\dfrac{{3\sqrt 3 {{\text{r}}^3}}}{8}} \right)}}{{{{\text{r}}^3}}} $
          = $ \dfrac{{3\sqrt 3 }}{4} $

Hence the ratio of the volumes of a solid sphere and a solid hemisphere is $ \dfrac{{3\sqrt 3 }}{4} $ .

Note - The key in solving such type of problems is to know the formulae of the surface areas and volumes of sphere and hemisphere respectively. Using this formulae we build a relation between their respective radii which is later used to determine the ratio of their volumes.