Question

Three solid spheres of radius $1\text{cm},6\text{cm}$ and $8\text{cm}$, respectively are melted together and cast into a single sphere. The radius $r$ of this sphere will be

• A. $r<9\text{cm}$
• B. $r=9\text{cm}$
• C. $9\text{cm}<r<10\text{cm}$
• D. $r>10\text{cm}$

Answer 1

Answer: (B)

$r=9\text{cm}$

Let $r_1=1\text{cm},r_2=6\text{cm}$ and $r_3=8\text{cm}.$

According to the question,

The volume of three spheres $=$ The volume of the new sphere

$\begin{array}{cc}\frac{4}{3}\pi {r_1}^3+\frac{4}{3}\pi {r_2}^3+\frac{4}{3}\pi {r_3}^3& =\frac{4}{3}\pi r^3\\ {\left(1\right)}^3+{\left(6\right)}^3+{\left(8\right)}^3& =r^3\\ 1+216+512& =r^3\\ r^3& =729\\ r^3& ={\left(9\right)}^3\\ r& =9\text{cm}\end{array}$

Hence, the radius of the resulting sphere is $9\text{cm}.$