Question

A spherical ball of radius 3 cm is melted and recast in three spherical ball the radius of two of the balls are 1.5cm and 2cm respectively determine the third ball

Answer 1

Given :

• A spherical ball of radius 3 cm is melted and recast in three spherical ball.
• Radius of the two of the balls formed is 1.5 cm and 2 cm, (respectively)

To Find :

• Radius of the third ball.

Solution :

Let the radius of the third spj

The bigger spherical ball of radius 3 cm is melted and 3 smalller spherical balls are formed.

Radius of the bigger ball is 3cm.

So while using the formula, volume of the larger spherical ball will be equal to the sum of the volume of the other three spherical ball.

Formula :

$\large{\boxed{\sf{\red{Volume_{sphere}\:=\:\dfrac{4}{3}\:\pi\:r^3}}}}$

Use this formula for all the 4 spherical balls. Since, when an object is melted and recasted, the volume of the initial object (shape) remains equal to the volume of the new objects/shapes formed.

$\sf{\dfrac{4}{3}\:\times\:\pi\:\times\:(3)^3\:=\:\dfrac{4}{3}\:\times\:\pi\:\times\:(1.5)^3\:+\:\dfrac{4}{3}\:\times\:\:\pi\:\:\times\:(2)^3\:+\:\dfrac{4}{3}\:\times\:\pi\:\:\times\:r^3}$

$\sf{\dfrac{4}{3}\:\times\:\pi\:\times\:27\:=\:\dfrac{4}{3}\:\times\:\pi\:\times\:3.375\:+\:\dfrac{4}{3}\:\times\:\pi\:\times\:8\:+\:\dfrac{4}{3}\:\times\:\pi\:\times\:r^3}$

$\sf{\dfrac{108\:\pi}{3}\:=\:\dfrac{4}{3}\:\pi\:(3.375\:+\:8\:+r^3)}$

$\sf{\dfrac{108\:\pi}{3}\:=\:\dfrac{4\:\pi}{3}\:(11.375+r^3)}$

$\sf{\frac{\frac{108\:\pi}{3}}{\frac{4\:\pi}{3}}\:=\:11.375+r^3}$

$\sf{\dfrac{108\:\pi}{3}\:\times\:\dfrac{3}{4\:\pi}\:=\:11.375\:+\:r^3}$

$\sf{\dfrac{108\:\cancel{\pi}}{3}\:\times\:\dfrac{3}{4\:\cancel{\pi}}\:=\:11.375\:+\:r^3}$

$\sf{\dfrac{\cancel{108}}{3}\:\times\:\dfrac{3}{\cancel{4}}\:=\:11.375\:+\:r^3}$

$\sf{\dfrac{27}{\cancel{3}}\:\times\:\cancel{3}\:=\:11.375+r^3}$

$\sf{27\:=\:11.375+r^3}$

$\sf{27-11.375=r^3}$

$\sf{15.625=r^3}$

$\sf{r\:=\:^3\sqrt{15.625}}$

$\sf{r=2.5}$

$\large{\boxed{\sf{\purple{Radius\:of\:third\:ball\:=\:2.5\:cm}}}}$

Answer 2

$\huge\underline\mathrm\purple{→Given←}$

• A spherical ball of radius 3cm
• After recast in three spherical ball the radius of balls are 1.5cm & 2cm

$\huge\underline\mathrm\purple{→To\:find←}$

• Determine the radius of third ball

$\large\underline\bold\orange{Note:}$

$\textbf{Volume\:of\:sphere}$

$\implies\large\sf \frac{4}{3}πr^3$

$\huge\underline\mathrm\purple{→Solution←}$

Let the radius of three spherical ball after recast be x , y and z

★The volume of larger spherical ball = Sum of Volume of three spherical ball

$\implies\sf \frac{4}{3}πR^3=\frac{4}{3}πx^3+\frac{4}{3}πy^3+\frac{4}{3}πz^3$

$\implies\sf \frac{4}{3}π(3)^3=\frac{4}{3}π(x^3+y^3+z^3)$

$\implies\sf \cancel\frac{4}{3}×\cancel{π}×27=\cancel\frac{4}{3}×\cancel{π}((1.5)^3+(2)^3+z^3)$

$\implies\sf 27=(3.375+8+z^3)$

$\implies\sf 27=(11.375+z^3)$

$\implies\sf 27-11.375=z^3$

$\implies\sf 15.625=z^3$

$\implies\sf z=2.5cm$

$\large{\boxed{\bf{\red{Radius\:of\:third\:ball\:=2.5cm}}}}$