Answer: Difference = 3x² + 3x + 1
Step-by-step explanation:
Required: How changes in sides of a cube affects its volume.
Take for instance the side of the cube is x.
The initial volume would be:
Volume = x * x * x
Volume = x³
When then dimension is increased by 1 unit, the new volume would be
Volume = (x + 1) * (x + 1) * (x + 1)
Expand the brackets
New Volume = (x² + 2x + 1)(x + 1)
New Volume = x³ + 3x² + 3x + 1
[Calculate the difference between both volumes]
Difference = New Volume - Initial Volume
Difference = x³ + 3x² + 3x + 1 - x³
[Collect like terms]
Difference = x³ - x³ + 3x² + 3x + 1
Difference = 3x² + 3x + 1
So, there will be a difference of 3x² + 3x + 1 when the dimension is increased from x to x + 1
Take for instance: a dimension of 2 units is increased to 3 units
Initial Volume = 2³ = 8
New Volume = 3³ = 27
Difference = 27 - 8
Difference = 19
Using the derived formula (x = 2)
Difference = 3x² + 3x + 1
Substitute 2 for x
Difference = 3 * 2² + 3 * 2 + 1
Difference = 3 * 4 + 6 + 1
Difference = 12 + 6 + 1
Difference = 19
