Question

A sphere is melted and half of the molten liquid is used to form $11$ identical cubes, whereas the remaining half is used to form $7$ identical smaller spheres. The ratio of the side of the cube to the radius of the new small sphere is -

• A. ${\left(\frac{4}{3}\right)}^{\frac{1}{3}}$
• B. ${\left(\frac{8}{3}\right)}^{\frac{1}{3}}$
• C. $(3{)}^{\frac{1}{3}}$
• D. $2$

Answer 1

The correct option is B ${\left(\frac{8}{3}\right)}^{\frac{1}{3}}$

Volume of $11$ small cubes $=$ Volume of $7$ small spheres

$11\times a^3=7\times \frac{4}{3}\times \left(\pi \right)\times r^3$

$(\frac{a}{r}{)}^3=7\times \frac{4}{3}\times \frac{22}{7}\times \frac{1}{11}$

$(\frac{a}{r}{)}^3=\frac{8}{3}$

$\frac{a}{r}={\left(\frac{8}{3}\right)}^{\frac{1}{3}}$