Question

A spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of two of these balls are 1.5 cm and 2 cm. Find the radius of the third ball.

Answer 1

Given that radius of the spherical ball r = 3 cm.

We know that the volume of the sphere $=\frac{4}{3}\pi r^3=\frac{4}{3}\times \frac{22}{7}\times (3{)}^3=\frac{4}{3}\times \frac{22}{7}\times 27=\frac{4\times 22\times 9}{7}=\frac{792}{7}=113.1428c{m}^3$

Given radii of the 1st sphere = 1.5 cm

Volume of the 1st sphere $=\frac{4}{3}\pi r^3=\frac{4}{3}\times \frac{22}{7}\times (1.5{)}^3=\frac{4}{3}\times \frac{22}{7}\times 3.375=\mathrm{14.1428.}$

Given radii of the 2nd sphere = 2 cm.

the volume of the 2nd sphere $=\frac{4}{3}\pi r^3=\frac{4}{3}\times \frac{22}{7}\times (2{)}^3=\mathrm{33.5238.}$

Volume of the two small spheres = 14.1428 + 33.5238

= 47.6666.

Volume of the third sphere = 113.1428 - 47.6666

= 65.4762 $c{m}^3$.

Let the radius of the third be r cm.

$\frac{4}{3}\pi r^3=65.4762\frac{4}{3}\times \frac{22}{7}\times r^3=65.4762r^3=65.4762\times \frac{7}{22}\times \frac{3}{4}=65.4762\times \frac{21}{88}=\frac{1375.0002}{88}=15.62500r^3=15.62500r=\sqrt[3]{315.62500}=2.5cm$

Therefore the radius of the third ball = 2.5 cm