Question

A cube of side 42 cm contains a sphere touching its sides. Find the volume in cubic cm of the gap in
between?

Answer 1

$\bf \: \underline{Given} :$ ↴

$\sf \: A \: cube \: of \: side \: 42 \: cm \: contains \: a \: sphere \: touching \: its \: sides.$

$\bf \: \underline{To \: find} :$ ↴

$\sf \: Find \: the \: volume \: in \: cubic \: cm \: of \: the \: gap \: in \: between ?$

$\bf \: \underline{Formula \: to \: be \: used} :$ ↴

$\implies \sf Volume \: of \: cube = (a)^{3}$

$\implies \sf Volume \: of \: sphere = \dfrac{4}{3} \pi r^{3}$

$\bf \: \underline{\underline{Solution}}$

To find the volume of gap between cube and sphere, first of all we need to find the volume of both, cube and sphere.

$\sf \: Finding \: the \: volume \: of \: cube,$

$\implies \sf Volume \: of \: cube = (a)^{3}$

$\implies \sf Volume \: of \: cube = (42)^{3}$

$\red{\sf \implies Volume \: of \: cube = 74088 \: cm^{3} }$

$\sf \: Now, finding \: the \: volume \: of \: sphere,$

$\sf \: Before \: putting \: the \: values, we \: need \: to \: find \: the \: radius.$

$\sf\implies Diameter \: of \: the \: sphere \: (2r) = 42 \: cm$

$\sf\implies Radius \: of \: the \: sphere \: (r) = \dfrac{42}{2}$

$\red{ \sf\implies Radius \: of \: the \: sphere = 21 \: cm}$

$\bf \: \underline{Now},$

$\implies \sf Volume \: of \: sphere = \dfrac{4}{3} \pi r^{3}$

$\implies \sf \dfrac{4}{3} \times \dfrac{22}{7} \times 21^{3}$

$\implies \sf \dfrac{4}{3} \times \dfrac{22}{7} \times 9261$

$\implies \sf \dfrac{88}{21} \times 9261$

$\implies \sf 88 \times 441$

$\implies \sf 38808$

$\red{ \sf \implies Volume \: of \: sphere = 38808 \: cm^{3} }$

Now, we have to subtract the volume of cube and volume of sphere to find the volume of gap between them.

$\sf \implies 74088 \: cm^{3} - 38808 \: cm^{3}$

$\red{\sf \implies 35280 \: cm^{3}}$

$\underline {\boxed{ \pink{ \sf \: Hence, volume \: of \: gap \: between \: them \: is \: 35280 \: cm ^{3}}}}$