Question

The surface area of a solid metallic sphere is 616cmsquare. It melted and recast into small sphere of diameter 3.5cm. how many such spheres can be obtained

Answer 1

$\sf\red{\underline{\underline{Answer:}}}$

$\sf{64 \ spheres \ can \ be \ obtained.}$

$\sf\orange{Given:}$

$\sf{For \ sphere(1),}$

$\sf{\implies{Surface \ area=616 \ cm^{2}}}$

$\sf{For \ sphere(2),}$

$\sf{\implies{Diameter (D)=3.5 \ cm}}$

$\sf\pink{To \ find:}$

$\sf{Number \ of \ spheres \ that \ can \ be \ obtained.}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{For \ sphere(1),}$

$\boxed{\sf{Surface \ area \ of \ sphere=4\pi\times \ r^{2}}}$

$\sf{\therefore{616=4\times\frac{22}{7}\times \ r^{2}}}$

$\sf{\therefore{r^{2}=\frac{616\times7}{4\times22}}}$

$\sf{\therefore{r^{2}=7\times7}}$

$\sf{On \ taking \ square \ root \ of \ both \ sides}$

$\sf{\therefore{r=7 \ cm}}$

$\sf{For \ sphere(2),}$

$\sf{Radius (R)=\frac{Diameter}{2}=\frac{3.5}{2}}$

$\sf{\therefore{Radius (R)=\frac{3.5}{2} \ cm}}$

$\boxed{\sf{Volume \ of \ Sphere=\frac{4}{3}\times\pi\times \ r^{3}}}$

$\sf{Number \ of \ sphere=\frac{Volume \ of \ sphere(1)}{Volume \ of \ sphere (2)}}$

$\sf{=\frac{\frac{4}{3}\times\pi\times \ r^{3}}{\frac{4}{3}\times\pi\times \ R^{3}}}$

$\sf{=\frac{r^{3}}{R^{3}}}$

$\sf{=\frac{7\times7\times7}{\frac{3.5}{2}\times\frac{3.5}{2}\times\frac{3.5}{2}}}$

$\sf{=\frac{7\times7\times7\times2\times2\times2}{3.5\times3.5\times3.5}}$

$\sf{=2\times2\times2\times2\times2\times2}$

$\sf{=8\times8}$

$\sf{\therefore{Number \ of \ sphere=64}}$

$\sf\purple{\tt{\therefore{64 \ spheres \ can \ be \ obtained.}}}$