When length of each side of a cube is increased by 3 cm, its volume is increased by 2457 cm³. Find its side. How much will its volume decrease, if length of each side of it is reduced by 20% ?

Let a be the side of the original cube.

Side of the new cube = a + 3

Volume of the new cube = a³ + 2457

⇒ a³ + 2457 = (a + 3)³

⇒ a³ + 2457 = a³ + 3³ + 3 x a² x 3 + 3 x a x 3²

⇒ $\cancel{a^3}$+ 2457 = $\cancel{a^3}$+ 27 + 9a² + 27a

⇒ 2457 = 27 + 9a² + 27a

⇒ 9a² + 27a - 2457 + 27 = 0

⇒ 9a² + 27a - 2430 = 0

⇒ a² + 3a - 270 = 0

⇒ a² + 18a - 15a - 270 = 0

⇒ a(a + 18) - 15(a + 18) = 0

⇒ (a + 18)(a - 15) = 0

⇒ a = -18 or 15

Since the side cannot be negative, the side of the original cube, a = 15 cm.

Volume of original cube = side³

= (15)³ cm³

= 3375 cm³

When the length of side is reduced by 20%.

New side of the cube = side - 20% of side

= 15 - $\dfrac{20}{100}$ x 15

= 15 - $\dfrac{1}{5}$ x 15

= 15 - 3

= 12 cm

New volume of the cube = side³

= (12)³ cm³

= 1728 cm³

Decrease in volume = 3375 - 1728 cm³

= 1647 cm³

Hence, the side of the original cube is 15 cm and the decrease in volume is 1647 cm³.