Question

When each side of a cube was increased by 2 cm the volume increased by 1016 cm³. Find the side of the cube. If each side is decreased by 2 cm, by how much will the volume decrease?

Answer 1

Given: When each side of a cube is increased by 2 cm the volume increases by 1016 cm³.

Step-wise calculation:

1. Let the side of the cube be x cm.

2. Original volume = x³.

New volume = (x + 2)³.

3. Increase:

(x + 2)³ – x³

= 1016

4. Expand:

x³ + 6x² + 12x + 8 – x³

= 1016

⇒ 6x² + 12x + 8 = 1016

5. Simplify:

6x² + 12x – 1008 = 0

⇒ Divide by 6

⇒ x² + 2x – 168 = 0

6. Solve quadratic:

Discriminant = 2² – 4(1)(–168) 

= 4 + 672

= 676 

= 26²

`x = [-2 ± 26]/2` 

⇒ x = `(24)/2` = 12 or x = `(-28)/2` = −14 (reject negative).

So x = 12 cm.

7. If each side is decreased by 2 cm, new side = 12 – 2 = 10 cm.

8. Volume decrease

= Original – Decreased

= 12³ – 10³

= 1728 – 1000

= 728 cm³

Alternatively use formula

x³ – (x – 2)³

= 6x² – 12x + 8

Plugging x = 12 gives 728.

The side of the cube is 12 cm. If each side is decreased by 2 cm the volume will decrease by 728 cm³.