What happens to the volume of a shape when the scale factor of enlargement is n?

What happens to the volume of a shape when the scale factor of enlargement is n?

• A. It increases by n
• B. It remains the same
• C. It increases by n²
• D. It increases by n³

Answer: (D)

When a shape undergoes enlargement with a scale factor of \( n \), every linear dimension of the shape, such as length, width, and height, is multiplied by \( n \). Volume is a three-dimensional measure, and it can be understood as the product of the length, width, and height of an object. Therefore, if each linear measurement is multiplied by \( n \), the new dimensions of the shape become \( n \times \text{length} \), \( n \times \text{width} \), and \( n \times \text{height} \). The volume of the original shape can be expressed as: \[ \text{Volume} = \text{length} \times \text{width} \times \text{height} \] For the enlarged shape, the new volume becomes: \[ \text{New Volume} = (n \times \text{length}) \times (n \times \text{width}) \times (n \times \text{height}) = n^3 \times (\text{length} \times \text{width} \times \text{height}) \] This shows that the new volume is \( n^3 \)

When a shape undergoes enlargement with a scale factor of ( n ), every linear dimension of the shape, such as length, width, and height, is multiplied by ( n ).

Volume is a three-dimensional measure, and it can be understood as the product of the length, width, and height of an object. Therefore, if each linear measurement is multiplied by ( n ), the new dimensions of the shape become ( n \times \text{length} ), ( n \times \text{width} ), and ( n \times \text{height} ).

The volume of the original shape can be expressed as:

[

\text{Volume} = \text{length} \times \text{width} \times \text{height}

]

For the enlarged shape, the new volume becomes:

[

\text{New Volume} = (n \times \text{length}) \times (n \times \text{width}) \times (n \times \text{height}) = n^3 \times (\text{length} \times \text{width} \times \text{height})

]

This shows that the new volume is ( n^3 )