Questions & Answers

Question

Answers

A.${\left( {\dfrac{4}{3}} \right)^{\dfrac{1}{3}}}$

B.${\left( {\dfrac{8}{3}} \right)^{\dfrac{1}{3}}}$

C.${\left( 3 \right)^{\dfrac{1}{3}}}$

D.$2$

Answer
Verified

Given,

Radius of the 7 spherical ball = r

The volume of the sphere = \[\dfrac{4}{3}{\text{ }}\pi {r^3}\]

As there are 7 spherical balls so total volume =\[7\left( {\dfrac{4}{3}{\text{ }}\pi {r^3}} \right) = 7\left( {\dfrac{4}{3}{\text{ }}\left( {\dfrac{{22}}{7}} \right){r^3}} \right) = \dfrac{{88}}{3}{r^3}\]

Side of the identical cubes = a

So, the volume of the cube = ${a^3}$

As there are 11 identical cubes so total volume = $11{a^3}$

Here it is given 11 identical cubes are made from half sphere melted and similarly 7 identical small spheres are made from half sphere melted. So, both the volumes are equal.

Hence, we get the below equation: -

\[

\dfrac{{88}}{3}{r^3} = 11{a^3} \\

\Rightarrow \dfrac{{{a^3}}}{{{r^3}}} = \dfrac{8}{3} \\

\Rightarrow \dfrac{a}{r} = {\left( {\dfrac{8}{3}} \right)^{\dfrac{1}{3}}} \\

\]

Hence, the ratio of the side of the cube to the radius of the new small sphere is \[{\left( {\dfrac{8}{3}} \right)^{\dfrac{1}{3}}}\]