Question

The radii of two spheres $s_1$ and $s_2$ are in the ratio 4:3. If $s_1$ and $s_2$ are melted and converted into a new sphere, then the ratio of volumes of sphere $s_1$ and the new sphere is

• A. 64:91
• B. 27:91
• C. 27:64
• D. 4:7

Answer 1

The correct option is A 64:91

Let $r_1$ and $r_2$ be the radii of spheres $s_1$ and $s_2$ respectively.

$\therefore r_1:r_2=4:3$

$\Rightarrow \frac{r_1}{r_2}=\frac{4}{3}$

$\Rightarrow r_2=\frac{3r_1}{4}$ ...(i)

Let $r$ be the radius of the new sphere.

Now,

Volume of new sphere = Sum of the volume of spheres $s_1$ and $s_2$

$\therefore \frac{4}{3}\pi r^3=\frac{4}{3}\pi r_1^3+\frac{4}{3}\pi r_2^3$

$\Rightarrow r^3=r_1^3+r_2^3$

$\Rightarrow r^3=r_1^3+{\left(\frac{3r_1}{4}\right)}^3$ [From (i)]

$\Rightarrow r^3=r_1^3+\frac{27r_1^3}{64}$

$\Rightarrow r^3=\frac{91}{64}{r}_1^3$ ...(ii)

$\therefore$ Ratio of volume of $s_1$ to new sphere $=\frac{4}{3}\pi r_1^3:\frac{4}{3}\pi r^3$

$=r_1^3:r^3$

$=r_1^3:\frac{91}{64}{r}_1^3$ ...[From (ii)]

= 64 : 91

Hence, the correct answer is option (a).