Question

A sphere is melted and recast into a hemisphere. Show that the ratio of surface area of sphere to that of hemisphere is 4:3$\sqrt[3]{4}$ (4 : 3 cube root of 4).

Answer 1

$\textbf{Given:}$

$\textsf{A sphere is melted and recast into a hemisphere}$

$\mathsf{}$

$\textbf{To prove:}$

$\textsf{Ratio of surface area of sphere to that of hemisphere is}$

$\mathsf{4:3\,\sqrt[3]{4}}$

$\textbf{Solution:}$

$\textbf{Formula used:}$

$\boxed{\begin{minipage}{10cm} \\\textsf{Surface of a sphere of radius 'r' units}\;\mathsf{=\dfrac{4}{3}\pi\,r^3\;units^3}\\\\\textsf{Surface of a hemisphere of radius 'r' units}\;\mathsf{=\dfrac{2}{3}\pi\,r^3\;units^3}\\ \end{minipage}}$

$\mathsf{Let\;r_1\;and\;r_2\;}\;\textsf{be the radii of sphere and hemisphere respectively}$

$\textsf{As per given data,}$

$\mathsf{\dfrac{4}{3}\pi\,{r_1}^3=\dfrac{2}{3}\pi\,{r_2}^3}$

$\mathsf{2\,{r_1}^3={r_2}^3}$

$\mathsf{\dfrac{{r_1}^3}{{r_2}^3}=\dfrac{1}{2}}$

$\implies\boxed{\mathsf{\dfrac{r_1}{r_2}=\dfrac{1}{\sqrt[3]{2}}}}$

$\mathsf{Now}$

$\mathsf{\dfrac{\textsf{Surface area of sphere}}{\textsf{Surface area of hemi-sphere}}}$

$\mathsf{=\dfrac{4\,\pi\,{r_1}^2}{3\,\pi\,{r_1}^2}}$

$\mathsf{=\dfrac{4\,{r_1}^2}{3\,{r_1}^2}}$

$\mathsf{=\dfrac{4(1)^2}{3(\sqrt[3]{2})^2}}$

$\mathsf{=\dfrac{4}{3(2^\frac{1}{3})^2}}$

$\mathsf{=\dfrac{4}{3(2^2)^\frac{1}{3}}}$

$\mathsf{=\dfrac{4}{3(4)^\frac{1}{3}}}$

$\mathsf{=\dfrac{4}{3\,\sqrt[3]{4}}}$

$\textbf{Answer:}$

$\mathsf{Ratio\;of\;surface\;areas\;is\;4:3\,\sqrt[3]{4}}$

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Answer 2

Answer:

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Step-by-step explanation:

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