Question

A sphere is melted and recast into a hemisphere.

Show that the ratio of surface area of sphere to

that of hemisphere is 3 4:3^3√4​

Answer 1

Answer:

As the sphere is melted and recast into a hemisphere,

Volume of sphere = Volume of hemisphere

Let the radius of sphere be r and radius of hemisphere be R

Volume of sphere = $\frac{4}{3} \pi r^{3}$ = V

Volume of hemisphere = $\frac{2}{3} \pi R^{3}$ = V

⇒ $\frac{4}{3} \pi r^{3}$  =  $\frac{2}{3} \pi R^{3}$

⇒ $2r^{3}$ = $R^3$

⇒ $r\sqrt[3]{2}$ = $R$  --------------> ( 1 )

Now,

      Surface area of sphere : Surface area of hemisphere

=     $4\pi r^{2}$ : $3\pi R^2$

=     $4r^2$ : $3R^2$

=     $4r^2$ : $3(r\sqrt[3]{2})^2$ [ From ( 1 ) ]

=     $4r^2$ : $3r^2(\sqrt[3]{2})^2$

=     $4$ : $3(\sqrt[3]{2^2})$

=     $4$ : $3(\sqrt[3]{4})$

Hence, proved.

Hope it helps!!! Please mark Brainliest!!!

Answer 2

It is super simple

CSA of Sphere: CSA of hemisphere = 3 4:3^3√4