Question

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

• A. $r<9cm$
• B. $r=9cm$
• C. $9cm<r<10cm$
• D. $r>10cm$

Answer 1

The correct option is B

$r=9cm$

Step 1: Find the total volume of three solid spheres.

Let the radius of the first sphere $r_1=1cm$,

the radius of the second sphere $r_2=6cm$ and

the radius of the third sphere $r_3=8cm$ and

Therefore, the total volume of all the spheres $=\frac{4}{3}{\mathrm{\pi r}}_1^3+\frac{4}{3}{\mathrm{\pi r}}_2^3+\frac{4}{3}{\mathrm{\pi r}}_3^3$

$$\begin{gathered} =\frac{4}{3}\mathrm{\pi }{\left(1\right)}^3+\frac{4}{3}\mathrm{\pi }{\left(6\right)}^3+\frac{4}{3}\mathrm{\pi }{\left(8\right)}^3 \\ =\frac{4}{3}\mathrm{\pi }+\frac{4}{3}\mathrm{\pi }\times 216+\frac{4}{3}\mathrm{\pi }\times 512 \\ =\frac{4}{3}\mathrm{\pi }\left(1+216+512\right){\mathrm{cm}}^3 \\ =\frac{4}{3}\mathrm{\pi }\times 729{\mathrm{cm}}^3 \end{gathered}$$

Step 2: Find the radius of the sphere

The volume of the sphere of radius $r$ $=\frac{4}{3}{\mathrm{\pi r}}^3$

According to the question, the total volume of all the spheres $=$ volume of a new sphere

$$\begin{gathered} \therefore \frac{4}{3}{\mathrm{\pi r}}^3=\frac{4}{3}\mathrm{\pi }\times 729 \\ \Rightarrow r^3=729 \\ \Rightarrow r=9cm \end{gathered}$$

Hence, radius $r$ of this sphere is $9cm$.

Therefore, the correct answer is option B.