Question

Aglass sphere with diameter 3 cm is melted and recasted into 3 smallsphere.If diamter of 2 spheres recasted are 1.5cmand 2cm
respectively find diameter
of third sphere.​

Answer 1

Given :

•A glass sphere with diameter =3cm

•The diameter of spheres recasted =1.5 cm and 2 cm

To Find :

• The diameter of third sphere =?

Formula:

$\\ \red \bigstar\boxed{ \sf{volume \: of \: sphere = \frac{4}{3} \pi {r}^{3} }}$

Solution :

Let the radius of third bail be r

⇒The volume of given largest sphere =$\sf{\dfrac{4}{3} \pi \bigg(\dfrac{3}{2} \bigg)^{3} }$

⇒The volume of the sphere of diameter 1.5cm =$\sf{\dfrac{4}{3} \pi \bigg(\dfrac{1.5}{2} \bigg)^{3} cm^{3} }$

⇒The volume of the sphere of diameter 2 cm=$\sf{\dfrac{4}{3} \pi \bigg(\dfrac{2}{3} \bigg)^{3} }$

⇒The volume of the sphere of radius r=$\sf{\dfrac{4}{3} \pi (r)^{3}}$

$\\ \green{\underline{\sf{According \: to \: given \: conditions }}}$

• Volume of the largest sphere =sum of the volume of sphere recasted

$\\ \implies \sf \: \frac{4}{3} \pi \bigg({ \frac{3}{2} } \bigg)^{3} = \frac{4}{3} \pi \bigg( { \frac{1.5}{2} } \bigg)^{3} + \frac{4}{3} \pi {(1)}^{3} + \frac{4}{3} \pi {r}^{3} \\ \\ \\ \implies \sf \: \: \bigg({ \frac{3}{2} } \bigg)^{3} = \bigg({ \frac{1.5}{2} } \bigg)^{3} + 1 + {r}^{3} \\ \\ \\ \implies \sf \: \frac{27}{8} = \frac{3.375}{8} + 1 + {4}^{3} \\ \\ \\ \implies \sf \: \frac{27}{8} = \frac{27}{64} + 1 + {r}^{3} \\ \\ \\ \implies \sf \: {r}^{3} = \frac{27}{8} - \frac{27}{64} - 1 \\ \\ \\ \implies \sf \: {r}^{3} = \frac{216 - 27 - 64}{64} \\ \\ \\ \implies \sf \: {r}^{3} = \frac{216 - 91}{64} \\ \\ \\ \implies \sf \: {r}^{3} = \frac{125}{64} \\ \\ \\ \implies \sf \: \: {r}^{3} = \bigg( { \frac{5}{4} } \bigg)^{3} \\ \\ \\ \implies \boxed{ \sf{r = \frac{5}{4} }} \:$

The diameter of third sphere is =2r

$\\ \\ \implies \sf \: diameter \: = 2r \\ \\ \\ \implies \sf \: diameter = 2 \times \frac{5}{4} \\ \\ \\ \implies \sf diameter = \frac{5}{2} \\ \\ \\ \implies \boxed{ \sf{diameter \: of \: third \: sphere = 2.5 \: cm}} \bigstar$