Explain how changes in the dimensions of a cube dimensions affect the volume of a cube. Be specific, explaining how much the volume will change with each increase of 1 unit on the side lengths.

  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

for ex: the reciprocal of 3/4 is 4/3. it is the opposite.

reciprocal definition:

a mathematical expression or function so related to another that their product is one; the quantity obtained by dividing the number one by a given quantity.

(x, y)→(x − 8, y − 7)

step-by-step explanation:

enjoy!