High School

Pentagonal prism B is the image of pentagonal prism A after dilation by a scale factor of 2. If the volume of pentagonal prism A is 56 in\(^3\), find the volume of pentagonal prism B, the image.

Answer :

The volume of pentagonal prism B, the image, is 448 cubic in.

The volume of a prism is directly proportional to the cube of the scale factor. Therefore, if prism B is the image of prism A after dilation by a scale factor of 2, the volume of prism B will be 2³ = 8 times the volume of prism A.

Given that the volume of prism A is 56 in³. Then the volume of prism B is calculated as,

=> Volume of prism A × (Scale factor)³

=> 56 in³ × 8

=> 448 in³

More about the dilation link is given below.

https://brainly.com/question/2856466

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The volume of pentagonal prism B is 448 [tex]in^{3}[/tex].

Firstly, when a three-dimensional shape is dilated by a scale factor of k, the volume of the resulting shape changes by a factor of [tex]k^{3}[/tex]. This is because volume is a three-dimensional measure.

Given:

The scale factor, k=2

The volume of pentagonal prism A, [tex]V_{A}[/tex]​=56 [tex]in^{3}[/tex]

  • Using the formula for the volume change:
    [tex]V_{B}[/tex] = [tex]k^{3}[/tex] × [tex]V_{A}[/tex]
    Where [tex]V_{B}[/tex]​ is the volume of prism B.
  • Substitute the values into the formula:
    [tex]V_{B}[/tex]​=[tex]2^{3}[/tex]×56

[tex]V_{B}[/tex]=8×56

[tex]V_{B}[/tex]=448 [tex]in^{3}[/tex]