When length of each side of a cube is increased by 3 cm, its volume is increased by 2457 $c{m}^3$. Find its side. How much will its volume decrease, if length of each side of it is reduced by 20%?

The correct option is D 15 cm, 1647 $c{m}^3$
Let the side of cube be 'a'. So, volume will be $a^3$.
Increased side of the cube a' = a + 3. Hence the increased volume = $a^3+2457$.
Increased volume of cube =
$a^3+2457=a^3+27+9a(a+3)\left[\right(a+b{)}^3=a^3+b^3+3aba+b\left)\right]⟹2457=27+9a^2+27a⟹9a^2+27a-2430=0⟹a^2+3a-270=0⟹a^2+18a-15a-270=0⟹a(a+18)-15(a+18)=0⟹a=15cm$
If the side was decreased by 20%, then side ${}^{\prime }{a}^{\prime }=12cm.$ The corresponding volume will be $12\ast 12\ast 12=1728$ cu. cm
Decrease $=15\ast 15\ast 15-12\ast 12\ast 12=3375-1728=1647$ cu. cm