Question

If a metallic hemisphere is recast into a sphere, then ratios of old radius to new radius are __________.

Answer 1

Find the volume of the metallic hemisphere and metallic sphere.

Let the radius of the hemisphere be $r_1$

The volume of the metallic hemisphere of radius $r_1$ $=\frac{2}{3}\pi {\left(r_1\right)}^3$

Let the radius of the sphere be $r_2$

The volume of the spherical of radius $r_2$ $=\frac{4}{3}\pi {\left(r_2\right)}^3$

Find the ratios of old radius to the new radius

We have, the volume of the metallic sphere $=$ Volume of the metallic hemispherical

$$\begin{gathered} \therefore \frac{2}{3}\pi {\left(r_1\right)}^3=\frac{4}{3}\pi {\left(r_2\right)}^3 \\ \Rightarrow {\left(r_1\right)}^3=2{\left(r_2\right)}^3 \\ \Rightarrow \frac{{\left(r_1\right)}^3}{{\left(r_2\right)}^3}=\frac{2}{1} \\ \Rightarrow \frac{r_1}{r_2}=\frac{\sqrt[3]{2}}{1} \\ \Rightarrow r_{1:}{r}_2=\sqrt[3]{2}:1 \end{gathered}$$

Hence, If a metallic hemisphere is recast into a sphere, then ratios of old radius to new radius are $\sqrt[3]{2}:1$