Question

A pentagonal prism with volume 2100cm^(3) is dilated with a factor of (1)/(2). What is the volume of the image?

Answer 1

The volume of the image of a pentagonal prism dilated with a factor of ½ is found by cubing the dilation factor and multiplying by the original volume, which results in a new volume of 262.5 cm³.

Step-by-step explanation:

The student asked about the change in volume of a pentagonal prism when it is dilated with a factor of ½. To find the volume of the image after dilation, we use the principle that the change in volume is proportional to the cube of the dilation factor. Since the original volume is 2100 cm³, and the dilation factor is ½, we calculate the new volume by raising the dilation factor to the third power ((½)³) and multiplying by the original volume.

Here are the steps:

1. Calculate the cube of the dilation factor: (½)³ = ¼ × ¼ × ¼ = ⅛.
2. Multiply this by the original volume: (⅛) × 2100 cm³ = 262.5 cm³.

Therefore, after dilation with a factor of ½, the new volume of the pentagonal prism will be 262.5 cm³.